Termination w.r.t. Q proof of /tmp/tmpTWueJI/PALINDROME_complete

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

U71¹(mark(X1), X2) → U71¹(X1, X2)
MARK(U11(X1, X2)) → MARK(X1)
ISNEPAL(mark(X)) → ISNEPAL(X)
ACTIVE(U62(tt)) → MARK(tt)
ACTIVE(U21(tt, V1, V2)) → ISLIST(V1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(isPalListKind(u)) → MARK(tt)
ACTIVE(U42(tt, V2)) → ISNELIST(V2)
U11¹(X1, mark(X2)) → U11¹(X1, X2)
ACTIVE(U23(tt)) → MARK(tt)
ACTIVE(isList(V)) → ISPALLISTKIND(V)
ACTIVE(isNePal(V)) → U61¹(isPalListKind(V), V)
U12¹(active(X)) → U12¹(X)
ACTIVE(U52(tt, V2)) → ISLIST(V2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(X1, mark(X2)) → U31¹(X1, X2)
ACTIVE(isList(__(V1, V2))) → ISPALLISTKIND(V1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(a) → ACTIVE(a)
ISPAL(active(X)) → ISPAL(X)
ISPALLISTKIND(mark(X)) → ISPALLISTKIND(X)
ACTIVE(isNePal(V)) → ISPALLISTKIND(V)
ACTIVE(U42(tt, V2)) → U43¹(isNeList(V2))
__¹(mark(X1), X2) → __¹(X1, X2)
ISLIST(mark(X)) → ISLIST(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
ACTIVE(isPal(V)) → ISPALLISTKIND(V)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
ACTIVE(isList(__(V1, V2))) → AND(isPalListKind(V1), isPalListKind(V2))
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
U72¹(active(X)) → U72¹(X)
MARK(U11(X1, X2)) → U11¹(mark(X1), X2)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(U31(X1, X2)) → U31¹(mark(X1), X2)
ACTIVE(U61(tt, V)) → ISQID(V)
U72¹(mark(X)) → U72¹(X)
U61¹(X1, active(X2)) → U61¹(X1, X2)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
ISNELIST(mark(X)) → ISNELIST(X)
ACTIVE(isNePal(__(I, __(P, I)))) → AND(isQid(I), isPalListKind(I))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(isNeList(V)) → U31¹(isPalListKind(V), V)
ACTIVE(isNeList(__(V1, V2))) → MARK(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ISPALLISTKIND(active(X)) → ISPALLISTKIND(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → U61¹(mark(X1), X2)
ACTIVE(isPalListKind(o)) → MARK(tt)
MARK(U43(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U12(tt)) → MARK(tt)
U42¹(X1, mark(X2)) → U42¹(X1, X2)
ACTIVE(isPal(V)) → U71¹(isPalListKind(V), V)
MARK(__(X1, X2)) → MARK(X2)
ACTIVE(isPalListKind(__(V1, V2))) → AND(isPalListKind(V1), isPalListKind(V2))
ACTIVE(isNeList(__(V1, V2))) → U41¹(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)
U12¹(mark(X)) → U12¹(X)
U22¹(X1, active(X2)) → U22¹(X1, X2)
U23¹(mark(X)) → U23¹(X)
__¹(X1, active(X2)) → __¹(X1, X2)
U43¹(mark(X)) → U43¹(X)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
MARK(U62(X)) → U62¹(mark(X))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
U61¹(X1, mark(X2)) → U61¹(X1, X2)
ACTIVE(isNeList(__(V1, V2))) → ISPALLISTKIND(V2)
MARK(i) → ACTIVE(i)
ACTIVE(isPalListKind(__(V1, V2))) → ISPALLISTKIND(V2)
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U43(X)) → ACTIVE(U43(mark(X)))
MARK(U51(X1, X2, X3)) → MARK(X1)
U53¹(active(X)) → U53¹(X)
ACTIVE(U51(tt, V1, V2)) → ISNELIST(V1)
ACTIVE(U31(tt, V)) → ISQID(V)
U31¹(X1, active(X2)) → U31¹(X1, X2)
MARK(U41(X1, X2, X3)) → MARK(X1)
U32¹(active(X)) → U32¹(X)
ACTIVE(isQid(u)) → MARK(tt)
AND(X1, active(X2)) → AND(X1, X2)
U21¹(X1, active(X2), X3) → U21¹(X1, X2, X3)
MARK(U12(X)) → ACTIVE(U12(mark(X)))
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(U51(X1, X2, X3)) → U51¹(mark(X1), X2, X3)
ISQID(mark(X)) → ISQID(X)
MARK(U53(X)) → U53¹(mark(X))
AND(active(X1), X2) → AND(X1, X2)
MARK(U72(X)) → ACTIVE(U72(mark(X)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U21(X1, X2, X3)) → U21¹(mark(X1), X2, X3)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
U52¹(X1, active(X2)) → U52¹(X1, X2)
ACTIVE(isQid(a)) → MARK(tt)
ACTIVE(isPal(nil)) → MARK(tt)
MARK(U42(X1, X2)) → U42¹(mark(X1), X2)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
U61¹(mark(X1), X2) → U61¹(X1, X2)
ACTIVE(U31(tt, V)) → U32¹(isQid(V))
ACTIVE(isQid(e)) → MARK(tt)
MARK(U12(X)) → U12¹(mark(X))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
MARK(and(X1, X2)) → AND(mark(X1), X2)
__¹(active(X1), X2) → __¹(X1, X2)
MARK(U53(X)) → ACTIVE(U53(mark(X)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
__¹(X1, mark(X2)) → __¹(X1, X2)
U42¹(X1, active(X2)) → U42¹(X1, X2)
ACTIVE(U11(tt, V)) → U12¹(isNeList(V))
ACTIVE(isNeList(__(V1, V2))) → AND(isPalListKind(V1), isPalListKind(V2))
ACTIVE(U21(tt, V1, V2)) → U22¹(isList(V1), V2)
MARK(U23(X)) → MARK(X)
ACTIVE(__(__(X, Y), Z)) → __¹(Y, Z)
U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
ACTIVE(isNePal(__(I, __(P, I)))) → AND(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(U41(tt, V1, V2)) → ISLIST(V1)
ACTIVE(U32(tt)) → MARK(tt)
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ISNELIST(active(X)) → ISNELIST(X)
U31¹(active(X1), X2) → U31¹(X1, X2)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
ACTIVE(isNeList(__(V1, V2))) → ISPALLISTKIND(V1)
U23¹(active(X)) → U23¹(X)
U52¹(active(X1), X2) → U52¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
ACTIVE(isNeList(V)) → ISPALLISTKIND(V)
ISQID(active(X)) → ISQID(X)
ACTIVE(isNeList(__(V1, V2))) → MARK(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(isList(__(V1, V2))) → U21¹(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)
U32¹(mark(X)) → U32¹(X)
U42¹(active(X1), X2) → U42¹(X1, X2)
ACTIVE(and(tt, X)) → MARK(X)
U61¹(active(X1), X2) → U61¹(X1, X2)
MARK(o) → ACTIVE(o)
MARK(U43(X)) → U43¹(mark(X))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isQid(X)) → ACTIVE(isQid(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
ACTIVE(U41(tt, V1, V2)) → U42¹(isList(V1), V2)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U53(tt)) → MARK(tt)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
ISPAL(mark(X)) → ISPAL(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
U11¹(active(X1), X2) → U11¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
ACTIVE(isNePal(__(I, __(P, I)))) → ISPALLISTKIND(I)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(__(__(X, Y), Z)) → __¹(X, __(Y, Z))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ISNEPAL(active(X)) → ISNEPAL(X)
MARK(U61(X1, X2)) → MARK(X1)
U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)
AND(mark(X1), X2) → AND(X1, X2)
ACTIVE(isList(nil)) → MARK(tt)
ACTIVE(U72(tt)) → MARK(tt)
U22¹(X1, mark(X2)) → U22¹(X1, X2)
U53¹(mark(X)) → U53¹(X)
ACTIVE(isNePal(__(I, __(P, I)))) → ISPAL(P)
U21¹(active(X1), X2, X3) → U21¹(X1, X2, X3)
MARK(tt) → ACTIVE(tt)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U32(X)) → U32¹(mark(X))
AND(X1, mark(X2)) → AND(X1, X2)
ACTIVE(__(X, nil)) → MARK(X)
ACTIVE(__(nil, X)) → MARK(X)
MARK(U22(X1, X2)) → U22¹(mark(X1), X2)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
MARK(U71(X1, X2)) → U71¹(mark(X1), X2)
ACTIVE(isPalListKind(nil)) → MARK(tt)
U42¹(mark(X1), X2) → U42¹(X1, X2)
ACTIVE(isNePal(__(I, __(P, I)))) → ISPALLISTKIND(P)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U53(X)) → MARK(X)
ACTIVE(isList(__(V1, V2))) → ISPALLISTKIND(V2)
ACTIVE(isList(__(V1, V2))) → MARK(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
MARK(U22(X1, X2)) → MARK(X1)
U43¹(active(X)) → U43¹(X)
MARK(U41(X1, X2, X3)) → U41¹(mark(X1), X2, X3)
ACTIVE(isPalListKind(a)) → MARK(tt)
MARK(__(X1, X2)) → __¹(mark(X1), mark(X2))
ACTIVE(U22(tt, V2)) → ISLIST(V2)
ACTIVE(isQid(o)) → MARK(tt)
ACTIVE(isNePal(__(I, __(P, I)))) → AND(isPal(P), isPalListKind(P))
ACTIVE(U61(tt, V)) → U62¹(isQid(V))
ACTIVE(isNePal(__(I, __(P, I)))) → ISQID(I)
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(U11(tt, V)) → ISNELIST(V)
U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)
ACTIVE(U71(tt, V)) → ISNEPAL(V)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
U11¹(mark(X1), X2) → U11¹(X1, X2)
MARK(U42(X1, X2)) → MARK(X1)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)
ACTIVE(isPalListKind(e)) → MARK(tt)
MARK(U12(X)) → MARK(X)
ACTIVE(U71(tt, V)) → U72¹(isNePal(V))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(U52(tt, V2)) → U53¹(isList(V2))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
U71¹(X1, mark(X2)) → U71¹(X1, X2)
U52¹(X1, mark(X2)) → U52¹(X1, X2)
ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)
ACTIVE(isPalListKind(__(V1, V2))) → ISPALLISTKIND(V1)
U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
U62¹(mark(X)) → U62¹(X)
U62¹(active(X)) → U62¹(X)
ACTIVE(U43(tt)) → MARK(tt)
MARK(u) → ACTIVE(u)
ACTIVE(isPalListKind(i)) → MARK(tt)
ACTIVE(isQid(i)) → MARK(tt)
U22¹(active(X1), X2) → U22¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
MARK(e) → ACTIVE(e)
U21¹(X1, X2, mark(X3)) → U21¹(X1, X2, X3)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
ACTIVE(U22(tt, V2)) → U23¹(isList(V2))
MARK(U72(X)) → U72¹(mark(X))
ACTIVE(isNePal(__(I, __(P, I)))) → MARK(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
MARK(U52(X1, X2)) → U52¹(mark(X1), X2)
MARK(U23(X)) → U23¹(mark(X))
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
ACTIVE(isNeList(__(V1, V2))) → U51¹(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)
ACTIVE(isList(V)) → U11¹(isPalListKind(V), V)
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U51(tt, V1, V2)) → U52¹(isNeList(V1), V2)
ISLIST(active(X)) → ISLIST(X)
MARK(nil) → ACTIVE(nil)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

U71¹(mark(X1), X2) → U71¹(X1, X2)
MARK(U11(X1, X2)) → MARK(X1)
ISNEPAL(mark(X)) → ISNEPAL(X)
ACTIVE(U62(tt)) → MARK(tt)
ACTIVE(U21(tt, V1, V2)) → ISLIST(V1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(isPalListKind(u)) → MARK(tt)
ACTIVE(U42(tt, V2)) → ISNELIST(V2)
U11¹(X1, mark(X2)) → U11¹(X1, X2)
ACTIVE(U23(tt)) → MARK(tt)
ACTIVE(isList(V)) → ISPALLISTKIND(V)
ACTIVE(isNePal(V)) → U61¹(isPalListKind(V), V)
U12¹(active(X)) → U12¹(X)
ACTIVE(U52(tt, V2)) → ISLIST(V2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(X1, mark(X2)) → U31¹(X1, X2)
ACTIVE(isList(__(V1, V2))) → ISPALLISTKIND(V1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(a) → ACTIVE(a)
ISPAL(active(X)) → ISPAL(X)
ISPALLISTKIND(mark(X)) → ISPALLISTKIND(X)
ACTIVE(isNePal(V)) → ISPALLISTKIND(V)
ACTIVE(U42(tt, V2)) → U43¹(isNeList(V2))
__¹(mark(X1), X2) → __¹(X1, X2)
ISLIST(mark(X)) → ISLIST(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
ACTIVE(isPal(V)) → ISPALLISTKIND(V)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
ACTIVE(isList(__(V1, V2))) → AND(isPalListKind(V1), isPalListKind(V2))
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
U72¹(active(X)) → U72¹(X)
MARK(U11(X1, X2)) → U11¹(mark(X1), X2)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(U31(X1, X2)) → U31¹(mark(X1), X2)
ACTIVE(U61(tt, V)) → ISQID(V)
U72¹(mark(X)) → U72¹(X)
U61¹(X1, active(X2)) → U61¹(X1, X2)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
ISNELIST(mark(X)) → ISNELIST(X)
ACTIVE(isNePal(__(I, __(P, I)))) → AND(isQid(I), isPalListKind(I))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(isNeList(V)) → U31¹(isPalListKind(V), V)
ACTIVE(isNeList(__(V1, V2))) → MARK(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ISPALLISTKIND(active(X)) → ISPALLISTKIND(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → U61¹(mark(X1), X2)
ACTIVE(isPalListKind(o)) → MARK(tt)
MARK(U43(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U12(tt)) → MARK(tt)
U42¹(X1, mark(X2)) → U42¹(X1, X2)
ACTIVE(isPal(V)) → U71¹(isPalListKind(V), V)
MARK(__(X1, X2)) → MARK(X2)
ACTIVE(isPalListKind(__(V1, V2))) → AND(isPalListKind(V1), isPalListKind(V2))
ACTIVE(isNeList(__(V1, V2))) → U41¹(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)
U12¹(mark(X)) → U12¹(X)
U22¹(X1, active(X2)) → U22¹(X1, X2)
U23¹(mark(X)) → U23¹(X)
__¹(X1, active(X2)) → __¹(X1, X2)
U43¹(mark(X)) → U43¹(X)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
MARK(U62(X)) → U62¹(mark(X))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
U61¹(X1, mark(X2)) → U61¹(X1, X2)
ACTIVE(isNeList(__(V1, V2))) → ISPALLISTKIND(V2)
MARK(i) → ACTIVE(i)
ACTIVE(isPalListKind(__(V1, V2))) → ISPALLISTKIND(V2)
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U43(X)) → ACTIVE(U43(mark(X)))
MARK(U51(X1, X2, X3)) → MARK(X1)
U53¹(active(X)) → U53¹(X)
ACTIVE(U51(tt, V1, V2)) → ISNELIST(V1)
ACTIVE(U31(tt, V)) → ISQID(V)
U31¹(X1, active(X2)) → U31¹(X1, X2)
MARK(U41(X1, X2, X3)) → MARK(X1)
U32¹(active(X)) → U32¹(X)
ACTIVE(isQid(u)) → MARK(tt)
AND(X1, active(X2)) → AND(X1, X2)
U21¹(X1, active(X2), X3) → U21¹(X1, X2, X3)
MARK(U12(X)) → ACTIVE(U12(mark(X)))
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(U51(X1, X2, X3)) → U51¹(mark(X1), X2, X3)
ISQID(mark(X)) → ISQID(X)
MARK(U53(X)) → U53¹(mark(X))
AND(active(X1), X2) → AND(X1, X2)
MARK(U72(X)) → ACTIVE(U72(mark(X)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U21(X1, X2, X3)) → U21¹(mark(X1), X2, X3)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
U52¹(X1, active(X2)) → U52¹(X1, X2)
ACTIVE(isQid(a)) → MARK(tt)
ACTIVE(isPal(nil)) → MARK(tt)
MARK(U42(X1, X2)) → U42¹(mark(X1), X2)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
U61¹(mark(X1), X2) → U61¹(X1, X2)
ACTIVE(U31(tt, V)) → U32¹(isQid(V))
ACTIVE(isQid(e)) → MARK(tt)
MARK(U12(X)) → U12¹(mark(X))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
MARK(and(X1, X2)) → AND(mark(X1), X2)
__¹(active(X1), X2) → __¹(X1, X2)
MARK(U53(X)) → ACTIVE(U53(mark(X)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
__¹(X1, mark(X2)) → __¹(X1, X2)
U42¹(X1, active(X2)) → U42¹(X1, X2)
ACTIVE(U11(tt, V)) → U12¹(isNeList(V))
ACTIVE(isNeList(__(V1, V2))) → AND(isPalListKind(V1), isPalListKind(V2))
ACTIVE(U21(tt, V1, V2)) → U22¹(isList(V1), V2)
MARK(U23(X)) → MARK(X)
ACTIVE(__(__(X, Y), Z)) → __¹(Y, Z)
U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
ACTIVE(isNePal(__(I, __(P, I)))) → AND(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(U41(tt, V1, V2)) → ISLIST(V1)
ACTIVE(U32(tt)) → MARK(tt)
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ISNELIST(active(X)) → ISNELIST(X)
U31¹(active(X1), X2) → U31¹(X1, X2)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
ACTIVE(isNeList(__(V1, V2))) → ISPALLISTKIND(V1)
U23¹(active(X)) → U23¹(X)
U52¹(active(X1), X2) → U52¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
ACTIVE(isNeList(V)) → ISPALLISTKIND(V)
ISQID(active(X)) → ISQID(X)
ACTIVE(isNeList(__(V1, V2))) → MARK(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(isList(__(V1, V2))) → U21¹(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)
U32¹(mark(X)) → U32¹(X)
U42¹(active(X1), X2) → U42¹(X1, X2)
ACTIVE(and(tt, X)) → MARK(X)
U61¹(active(X1), X2) → U61¹(X1, X2)
MARK(o) → ACTIVE(o)
MARK(U43(X)) → U43¹(mark(X))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isQid(X)) → ACTIVE(isQid(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
ACTIVE(U41(tt, V1, V2)) → U42¹(isList(V1), V2)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U53(tt)) → MARK(tt)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
ISPAL(mark(X)) → ISPAL(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
U11¹(active(X1), X2) → U11¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
ACTIVE(isNePal(__(I, __(P, I)))) → ISPALLISTKIND(I)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(__(__(X, Y), Z)) → __¹(X, __(Y, Z))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ISNEPAL(active(X)) → ISNEPAL(X)
MARK(U61(X1, X2)) → MARK(X1)
U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)
AND(mark(X1), X2) → AND(X1, X2)
ACTIVE(isList(nil)) → MARK(tt)
ACTIVE(U72(tt)) → MARK(tt)
U22¹(X1, mark(X2)) → U22¹(X1, X2)
U53¹(mark(X)) → U53¹(X)
ACTIVE(isNePal(__(I, __(P, I)))) → ISPAL(P)
U21¹(active(X1), X2, X3) → U21¹(X1, X2, X3)
MARK(tt) → ACTIVE(tt)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U32(X)) → U32¹(mark(X))
AND(X1, mark(X2)) → AND(X1, X2)
ACTIVE(__(X, nil)) → MARK(X)
ACTIVE(__(nil, X)) → MARK(X)
MARK(U22(X1, X2)) → U22¹(mark(X1), X2)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
MARK(U71(X1, X2)) → U71¹(mark(X1), X2)
ACTIVE(isPalListKind(nil)) → MARK(tt)
U42¹(mark(X1), X2) → U42¹(X1, X2)
ACTIVE(isNePal(__(I, __(P, I)))) → ISPALLISTKIND(P)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U53(X)) → MARK(X)
ACTIVE(isList(__(V1, V2))) → ISPALLISTKIND(V2)
ACTIVE(isList(__(V1, V2))) → MARK(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
MARK(U22(X1, X2)) → MARK(X1)
U43¹(active(X)) → U43¹(X)
MARK(U41(X1, X2, X3)) → U41¹(mark(X1), X2, X3)
ACTIVE(isPalListKind(a)) → MARK(tt)
MARK(__(X1, X2)) → __¹(mark(X1), mark(X2))
ACTIVE(U22(tt, V2)) → ISLIST(V2)
ACTIVE(isQid(o)) → MARK(tt)
ACTIVE(isNePal(__(I, __(P, I)))) → AND(isPal(P), isPalListKind(P))
ACTIVE(U61(tt, V)) → U62¹(isQid(V))
ACTIVE(isNePal(__(I, __(P, I)))) → ISQID(I)
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(U11(tt, V)) → ISNELIST(V)
U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)
ACTIVE(U71(tt, V)) → ISNEPAL(V)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
U11¹(mark(X1), X2) → U11¹(X1, X2)
MARK(U42(X1, X2)) → MARK(X1)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)
ACTIVE(isPalListKind(e)) → MARK(tt)
MARK(U12(X)) → MARK(X)
ACTIVE(U71(tt, V)) → U72¹(isNePal(V))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(U52(tt, V2)) → U53¹(isList(V2))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
U71¹(X1, mark(X2)) → U71¹(X1, X2)
U52¹(X1, mark(X2)) → U52¹(X1, X2)
ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)
ACTIVE(isPalListKind(__(V1, V2))) → ISPALLISTKIND(V1)
U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
U62¹(mark(X)) → U62¹(X)
U62¹(active(X)) → U62¹(X)
ACTIVE(U43(tt)) → MARK(tt)
MARK(u) → ACTIVE(u)
ACTIVE(isPalListKind(i)) → MARK(tt)
ACTIVE(isQid(i)) → MARK(tt)
U22¹(active(X1), X2) → U22¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
MARK(e) → ACTIVE(e)
U21¹(X1, X2, mark(X3)) → U21¹(X1, X2, X3)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
ACTIVE(U22(tt, V2)) → U23¹(isList(V2))
MARK(U72(X)) → U72¹(mark(X))
ACTIVE(isNePal(__(I, __(P, I)))) → MARK(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
MARK(U52(X1, X2)) → U52¹(mark(X1), X2)
MARK(U23(X)) → U23¹(mark(X))
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
ACTIVE(isNeList(__(V1, V2))) → U51¹(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)
ACTIVE(isList(V)) → U11¹(isPalListKind(V), V)
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U51(tt, V1, V2)) → U52¹(isNeList(V1), V2)
ISLIST(active(X)) → ISLIST(X)
MARK(nil) → ACTIVE(nil)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 26 SCCs with 95 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISPAL(mark(X)) → ISPAL(X)
ISPAL(active(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISPAL(mark(X)) → ISPAL(X)
ISPAL(active(X)) → ISPAL(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

ISPAL(mark(X)) → ISPAL(X)
The graph contains the following edges 1 > 1
ISPAL(active(X)) → ISPAL(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISPALLISTKIND(mark(X)) → ISPALLISTKIND(X)
ISPALLISTKIND(active(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISPALLISTKIND(mark(X)) → ISPALLISTKIND(X)
ISPALLISTKIND(active(X)) → ISPALLISTKIND(X)

From the DPs we obtained the following set of size-change graphs:

ISPALLISTKIND(mark(X)) → ISPALLISTKIND(X)
The graph contains the following edges 1 > 1
ISPALLISTKIND(active(X)) → ISPALLISTKIND(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(X1, active(X2)) → AND(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

From the DPs we obtained the following set of size-change graphs:

AND(mark(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(active(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(X1, mark(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
AND(X1, active(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNEPAL(mark(X)) → ISNEPAL(X)
ISNEPAL(active(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNEPAL(mark(X)) → ISNEPAL(X)
ISNEPAL(active(X)) → ISNEPAL(X)

From the DPs we obtained the following set of size-change graphs:

ISNEPAL(mark(X)) → ISNEPAL(X)
The graph contains the following edges 1 > 1
ISNEPAL(active(X)) → ISNEPAL(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U72¹(mark(X)) → U72¹(X)
U72¹(active(X)) → U72¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U72¹(mark(X)) → U72¹(X)
U72¹(active(X)) → U72¹(X)

From the DPs we obtained the following set of size-change graphs:

U72¹(mark(X)) → U72¹(X)
The graph contains the following edges 1 > 1
U72¹(active(X)) → U72¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U71¹(X1, active(X2)) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(X1, mark(X2)) → U71¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U71¹(X1, active(X2)) → U71¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U71¹(active(X1), X2) → U71¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U71¹(mark(X1), X2) → U71¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U71¹(X1, mark(X2)) → U71¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U62¹(mark(X)) → U62¹(X)
U62¹(active(X)) → U62¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U62¹(mark(X)) → U62¹(X)
U62¹(active(X)) → U62¹(X)

From the DPs we obtained the following set of size-change graphs:

U62¹(mark(X)) → U62¹(X)
The graph contains the following edges 1 > 1
U62¹(active(X)) → U62¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U61¹(X1, mark(X2)) → U61¹(X1, X2)
U61¹(X1, active(X2)) → U61¹(X1, X2)
U61¹(active(X1), 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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U61¹(X1, mark(X2)) → U61¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U61¹(X1, active(X2)) → U61¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U61¹(active(X1), X2) → U61¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U61¹(mark(X1), X2) → U61¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U53¹(mark(X)) → U53¹(X)
U53¹(active(X)) → U53¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U53¹(mark(X)) → U53¹(X)
U53¹(active(X)) → U53¹(X)

From the DPs we obtained the following set of size-change graphs:

U53¹(mark(X)) → U53¹(X)
The graph contains the following edges 1 > 1
U53¹(active(X)) → U53¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)
U52¹(X1, mark(X2)) → U52¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U52¹(active(X1), X2) → U52¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U52¹(mark(X1), X2) → U52¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U52¹(X1, active(X2)) → U52¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U52¹(X1, mark(X2)) → U52¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(active(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U43¹(active(X)) → U43¹(X)
U43¹(mark(X)) → U43¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U43¹(active(X)) → U43¹(X)
U43¹(mark(X)) → U43¹(X)

From the DPs we obtained the following set of size-change graphs:

U43¹(active(X)) → U43¹(X)
The graph contains the following edges 1 > 1
U43¹(mark(X)) → U43¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U42¹(mark(X1), X2) → U42¹(X1, X2)
U42¹(active(X1), X2) → U42¹(X1, X2)
U42¹(X1, mark(X2)) → U42¹(X1, X2)
U42¹(X1, active(X2)) → U42¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U42¹(mark(X1), X2) → U42¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U42¹(active(X1), X2) → U42¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U42¹(X1, mark(X2)) → U42¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U42¹(X1, active(X2)) → U42¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, 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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISQID(active(X)) → ISQID(X)
ISQID(mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISQID(active(X)) → ISQID(X)
ISQID(mark(X)) → ISQID(X)

From the DPs we obtained the following set of size-change graphs:

ISQID(active(X)) → ISQID(X)
The graph contains the following edges 1 > 1
ISQID(mark(X)) → ISQID(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U32¹(active(X)) → U32¹(X)
U32¹(mark(X)) → U32¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U32¹(active(X)) → U32¹(X)
U32¹(mark(X)) → U32¹(X)

From the DPs we obtained the following set of size-change graphs:

U32¹(active(X)) → U32¹(X)
The graph contains the following edges 1 > 1
U32¹(mark(X)) → U32¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U31¹(X1, active(X2)) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(X1, mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U31¹(X1, active(X2)) → U31¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U31¹(active(X1), X2) → U31¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U31¹(mark(X1), X2) → U31¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U31¹(X1, mark(X2)) → U31¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U23¹(mark(X)) → U23¹(X)
U23¹(active(X)) → U23¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U23¹(mark(X)) → U23¹(X)
U23¹(active(X)) → U23¹(X)

From the DPs we obtained the following set of size-change graphs:

U23¹(mark(X)) → U23¹(X)
The graph contains the following edges 1 > 1
U23¹(active(X)) → U23¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISLIST(mark(X)) → ISLIST(X)
ISLIST(active(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISLIST(active(X)) → ISLIST(X)
ISLIST(mark(X)) → ISLIST(X)

From the DPs we obtained the following set of size-change graphs:

ISLIST(mark(X)) → ISLIST(X)
The graph contains the following edges 1 > 1
ISLIST(active(X)) → ISLIST(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, mark(X2)) → U22¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U22¹(mark(X1), X2) → U22¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U22¹(X1, active(X2)) → U22¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U22¹(active(X1), X2) → U22¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U22¹(X1, mark(X2)) → U22¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U21¹(active(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)
U21¹(X1, X2, mark(X3)) → U21¹(X1, X2, X3)
U21¹(X1, active(X2), X3) → U21¹(X1, X2, X3)
U21¹(X1, mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U21¹(active(X1), X2, X3) → U21¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U21¹(X1, X2, mark(X3)) → U21¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U21¹(X1, active(X2), X3) → U21¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNELIST(active(X)) → ISNELIST(X)
ISNELIST(mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNELIST(active(X)) → ISNELIST(X)
ISNELIST(mark(X)) → ISNELIST(X)

From the DPs we obtained the following set of size-change graphs:

ISNELIST(active(X)) → ISNELIST(X)
The graph contains the following edges 1 > 1
ISNELIST(mark(X)) → ISNELIST(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U12¹(mark(X)) → U12¹(X)
U12¹(active(X)) → U12¹(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U12¹(mark(X)) → U12¹(X)
U12¹(active(X)) → U12¹(X)

From the DPs we obtained the following set of size-change graphs:

U12¹(mark(X)) → U12¹(X)
The graph contains the following edges 1 > 1
U12¹(active(X)) → U12¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)
U11¹(active(X1), 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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U11¹(X1, mark(X2)) → U11¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U11¹(X1, active(X2)) → U11¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U11¹(active(X1), X2) → U11¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U11¹(mark(X1), X2) → U11¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

__¹(X1, active(X2)) → __¹(X1, X2)
__¹(active(X1), X2) → __¹(X1, X2)
__¹(X1, mark(X2)) → __¹(X1, X2)
__¹(mark(X1), 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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

__¹(X1, active(X2)) → __¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
__¹(active(X1), X2) → __¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
__¹(X1, mark(X2)) → __¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
__¹(mark(X1), X2) → __¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(__(X, nil)) → MARK(X)
ACTIVE(__(nil, X)) → MARK(X)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U43(X)) → ACTIVE(U43(mark(X)))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
ACTIVE(isList(__(V1, V2))) → MARK(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U12(X)) → ACTIVE(U12(mark(X)))
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(isNeList(__(V1, V2))) → MARK(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U72(X)) → ACTIVE(U72(mark(X)))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(isQid(X)) → ACTIVE(isQid(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(isPal(X)) → ACTIVE(isPal(X))
MARK(U53(X)) → ACTIVE(U53(mark(X)))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(isNeList(__(V1, V2))) → MARK(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
ACTIVE(isNePal(__(I, __(P, I)))) → MARK(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
MARK(__(X1, X2)) → MARK(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U43(X)) → ACTIVE(U43(mark(X)))
MARK(U12(X)) → ACTIVE(U12(mark(X)))
MARK(U72(X)) → ACTIVE(U72(mark(X)))
MARK(U62(X)) → ACTIVE(U62(mark(X)))
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(isQid(X)) → ACTIVE(isQid(X))
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(U53(X)) → ACTIVE(U53(mark(X)))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(__(X, nil)) → MARK(X)
ACTIVE(__(nil, X)) → MARK(X)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
ACTIVE(isList(__(V1, V2))) → MARK(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(isNeList(__(V1, V2))) → MARK(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(isNeList(__(V1, V2))) → MARK(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
ACTIVE(isNePal(__(I, __(P, I)))) → MARK(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
MARK(__(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 1
POL(U22(x₁, x₂)) = 1
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 1
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 1
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 1
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 1
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(__(X, nil)) → MARK(X)
ACTIVE(__(nil, X)) → MARK(X)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
ACTIVE(isList(__(V1, V2))) → MARK(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
ACTIVE(isNeList(__(V1, V2))) → MARK(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(isNeList(__(V1, V2))) → MARK(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
ACTIVE(isNePal(__(I, __(P, I)))) → MARK(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
MARK(__(X1, X2)) → MARK(X2)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(__(X, nil)) → MARK(X)
ACTIVE(__(nil, X)) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
ACTIVE(isNePal(__(I, __(P, I)))) → MARK(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
MARK(__(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
ACTIVE(isList(__(V1, V2))) → MARK(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
ACTIVE(isNeList(__(V1, V2))) → MARK(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(isNeList(__(V1, V2))) → MARK(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁ + x₂
POL(U62(x₁)) = x₁
POL(U71(x₁, x₂)) = x₁ + x₂
POL(U72(x₁)) = x₁
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 1
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = x₁
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
active(U53(tt)) → mark(tt)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U62(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
ACTIVE(isList(__(V1, V2))) → MARK(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(isNeList(__(V1, V2))) → MARK(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(isNeList(__(V1, V2))) → MARK(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isList(__(V1, V2))) → MARK(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(isNeList(__(V1, V2))) → MARK(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
ACTIVE(isNeList(__(V1, V2))) → MARK(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U22(x₁, x₂)) = x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁ + x₂
POL(U62(x₁)) = x₁
POL(U71(x₁, x₂)) = x₁ + x₂
POL(U72(x₁)) = x₁
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = x₁
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
active(U53(tt)) → mark(tt)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U62(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNePal(V)) → MARK(U61(isPalListKind(V), V))
MARK(U71(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = x₁
POL(U71(x₁, x₂)) = 1 + x₁
POL(U72(x₁)) = x₁
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 1
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(U21(tt, V1, V2)) → MARK(U22(isList(V1), V2))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(isNePal(X)) → ACTIVE(isNePal(X))
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U22(x₁, x₂)) = x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = x₁
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = x₁
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

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

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U62(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNePal(X)) → ACTIVE(isNePal(X))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNeList(V1), V2))
MARK(U51(X1, X2, X3)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U62(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNePal(X)) → ACTIVE(isNePal(X))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = x₂ + x₃
POL(U22(x₁, x₂)) = x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = x₁
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = x₁
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

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

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNePal(X)) → ACTIVE(isNePal(X))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(isNePal(X)) → ACTIVE(isNePal(X))

The remaining pairs can at least be oriented weakly.

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 1
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 1
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 1
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

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

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U62(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U61(tt, V)) → MARK(U62(isQid(V)))
MARK(U61(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U62(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = 1 + x₁
POL(U62(x₁)) = x₁
POL(U71(x₁, x₂)) = 1
POL(U72(x₁)) = x₁
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 1
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

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

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 1
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 1
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

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

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U62(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U62(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = 1
POL(U62(x₁)) = 1 + x₁
POL(U71(x₁, x₂)) = 1
POL(U72(x₁)) = x₁
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 1
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 1
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

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

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U72(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 1 + x₂
POL(U72(x₁)) = 1 + x₁
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = 1 + x₁
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

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

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isPal(V)) → MARK(U71(isPalListKind(V), V))

The remaining pairs can at least be oriented weakly.

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 1
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 1
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

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

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U71(tt, V)) → MARK(U72(isNePal(V)))

The remaining pairs can at least be oriented weakly.

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = 1 + x₂
POL(U62(x₁)) = 1
POL(U71(x₁, x₂)) = 1
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

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

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isPal(X)) → ACTIVE(isPal(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(isPal(X)) → ACTIVE(isPal(X))

The remaining pairs can at least be oriented weakly.

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 1
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 1
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

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

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 1
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

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

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U41(X1, X2, X3)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U41(tt, V1, V2)) → MARK(U42(isList(V1), V2))
MARK(U41(X1, X2, X3)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U22(x₁, x₂)) = x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₂ + x₃
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁ + x₂
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = x₂
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = x₁
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

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

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))

The remaining pairs can at least be oriented weakly.

MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 1
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 1
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

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

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U22(X1, X2)) → MARK(X1)
ACTIVE(U22(tt, V2)) → MARK(U23(isList(V2)))

The remaining pairs can at least be oriented weakly.

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U22(x₁, x₂)) = 1 + x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 1

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

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

MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

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

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U23(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U22(x₁, x₂)) = 1 + x₂
POL(U23(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₂
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = 1 + x₁
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

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

MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U52(X1, X2)) → MARK(X1)
ACTIVE(isList(V)) → MARK(U11(isPalListKind(V), V))

The remaining pairs can at least be oriented weakly.

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = x₃
POL(U22(x₁, x₂)) = x₂
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U52(x₁, x₂)) = 1 + x₁ + x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1 + x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = x₁
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

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

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(isList(X)) → ACTIVE(isList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(isList(X)) → ACTIVE(isList(X))

The remaining pairs can at least be oriented weakly.

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isNeList(X)) → ACTIVE(isNeList(X))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
MARK(U53(X)) → MARK(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))
MARK(U53(X)) → MARK(X)

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(U12(x₁)) = 1 + x₁
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 1 + x₃
POL(U52(x₁, x₂)) = 1 + x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 1
POL(isList(x₁)) = 1 + x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 1
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U53(X)) → MARK(X)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(U52(tt, V2)) → MARK(U53(isList(V2)))
MARK(U42(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isQid(V)))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U53(X)) → MARK(X)

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U42(x₁, x₂)) = 1 + x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁ + x₂
POL(U62(x₁)) = x₁
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 1 + x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 1
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
active(U53(tt)) → mark(tt)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U53(X)) → MARK(X)

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U32(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U53(X)) → MARK(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U32(X)) → MARK(X)
MARK(U53(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(and(tt, X)) → MARK(X)

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 1 + x₁
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 1
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 1 + x₁
POL(U61(x₁, x₂)) = x₂
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = x₁
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
active(U53(tt)) → mark(tt)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNeList(V)) → MARK(U31(isPalListKind(V), V))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = x₁
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 1
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 1
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
active(U53(tt)) → mark(tt)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → ACTIVE(isNeList(X))
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(isNeList(X)) → ACTIVE(isNeList(X))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

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

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U11(tt, V)) → MARK(U12(isNeList(V)))

The remaining pairs can at least be oriented weakly.

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 1
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 1
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 1
POL(i) = 0
POL(isList(x₁)) = 1 + x₁
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
active(U53(tt)) → mark(tt)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

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

MARK(U43(X)) → MARK(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

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

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(and(tt, X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = 1
POL(U21(x₁, x₂, x₃)) = x₃
POL(U22(x₁, x₂)) = x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = 1
POL(U41(x₁, x₂, x₃)) = x₃
POL(U42(x₁, x₂)) = x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₃
POL(U52(x₁, x₂)) = x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = 1
POL(U62(x₁)) = 1
POL(U71(x₁, x₂)) = 1
POL(U72(x₁)) = 1
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 1
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 1
POL(i) = 1
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(o) = 1
POL(tt) = 1
POL(u) = 1

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
active(U53(tt)) → mark(tt)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

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

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 0
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

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

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 0
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

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

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U42(tt, V2)) → MARK(U43(isNeList(V2)))

The remaining pairs can at least be oriented weakly.

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1 + x₁ + x₂
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 0
POL(a) = 0
POL(active(x₁)) = 1
POL(and(x₁, x₂)) = 0
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(o) = 0
POL(tt) = 1
POL(u) = 0

The following usable rules [17] were oriented:

isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

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

MARK(U43(X)) → MARK(X)
ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U43(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 1 + x₁
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 0
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

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

ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isPalListKind(__(V1, V2))) → MARK(and(isPalListKind(V1), isPalListKind(V2)))

The remaining pairs can at least be oriented weakly.

MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1 + x₃
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = x₁
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
active(U53(tt)) → mark(tt)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

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

MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(isPalListKind(X)) → ACTIVE(isPalListKind(X))
MARK(and(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 0
POL(a) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1 + x₁
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 1
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

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

MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(__(X1, X2)) → ACTIVE(__(mark(X1), mark(X2)))
ACTIVE(__(__(X, Y), Z)) → MARK(__(X, __(Y, Z)))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
__(x1, x2) = __(x1, x2)
ACTIVE(x1) = x1
mark(x1) = x1
U51(x1, x2, x3) = U51(x1, x2)
active(x1) = x1
U43(x1) = U43(x1)
U42(x1, x2) = U42(x1, x2)
U32(x1) = x1
U41(x1, x2, x3) = U41(x1, x2, x3)
isQid(x1) = isQid
isList(x1) = x1
U31(x1, x2) = U31(x1)
U23(x1) = U23
U53(x1) = U53
tt = tt
isPalListKind(x1) = x1
isPal(x1) = isPal(x1)
U62(x1) = U62
U72(x1) = U72(x1)
nil = nil
isNePal(x1) = isNePal(x1)
and(x1, x2) = x2
U71(x1, x2) = U71(x1, x2)
U12(x1) = U12
U52(x1, x2) = x1
U61(x1, x2) = U61
a = a
o = o
u = u
i = i
e = e
isNeList(x1) = isNeList(x1)
U22(x1, x2) = U22
U21(x1, x2, x3) = U21(x1, x2, x3)
U11(x1, x2) = x1

Recursive path order with status [2].
Quasi-Precedence:

__₂ > MARK₁
__₂ > U41₃ > U42₂ > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > [U51₂, U31₁, isNeList₁]
__₂ > U41₃ > U42₂ > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > U72₁
__₂ > isPal₁ > U71₂ > isNePal₁ > [U62, U61] > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > [U51₂, U31₁, isNeList₁]
__₂ > isPal₁ > U71₂ > isNePal₁ > [U62, U61] > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > U72₁
__₂ > U21₃
a > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > [U51₂, U31₁, isNeList₁]
a > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > U72₁
o > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > [U51₂, U31₁, isNeList₁]
o > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > U72₁
u > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > [U51₂, U31₁, isNeList₁]
u > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > U72₁
e > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > [U51₂, U31₁, isNeList₁]
e > [U43₁, isQid, U23, U53, tt, nil, U12, i, U22] > U72₁

Status:

i: multiset
__₂: [1,2]
U22: []
U41₃: multiset
U71₂: [2,1]
U51₂: multiset
U61: multiset
tt: multiset
U72₁: multiset
U23: []
isQid: []
U21₃: multiset
nil: multiset
a: multiset
U62: multiset
U42₂: [1,2]
e: multiset
U53: []
U12: []
isNePal₁: [1]
o: multiset
U43₁: multiset
isPal₁: [1]
MARK₁: multiset
U31₁: multiset
u: multiset
isNeList₁: multiset

The following usable rules [17] were oriented:

U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
isQid(active(X)) → isQid(X)
isQid(mark(X)) → isQid(X)
isList(active(X)) → isList(X)
isList(mark(X)) → isList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
active(U53(tt)) → mark(tt)
isPalListKind(active(X)) → isPalListKind(X)
isPalListKind(mark(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)
active(U62(tt)) → mark(tt)
active(U72(tt)) → mark(tt)
active(isList(nil)) → mark(tt)
U72(active(X)) → U72(X)
U72(mark(X)) → U72(X)
isNePal(active(X)) → isNePal(X)
isNePal(mark(X)) → isNePal(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
active(isPal(nil)) → mark(tt)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
active(U12(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
active(U32(tt)) → mark(tt)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
active(U43(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(isQid(a)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
mark(tt) → active(tt)
mark(U62(X)) → active(U62(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt, V)) → mark(U12(isNeList(V)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isList(X)) → active(isList(X))
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
mark(isNeList(X)) → active(isNeList(X))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
mark(U53(X)) → active(U53(mark(X)))
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(U71(tt, V)) → mark(U72(isNePal(V)))
mark(U72(X)) → active(U72(mark(X)))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(isPal(X)) → active(isPal(X))
active(and(tt, X)) → mark(X)
mark(U12(X)) → active(U12(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
active(U31(tt, V)) → mark(U32(isQid(V)))
mark(isQid(X)) → active(isQid(X))
active(U61(tt, V)) → mark(U62(isQid(V)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
mark(U23(X)) → active(U23(mark(X)))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(nil) → active(nil)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
mark(a) → active(a)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ PisEmptyProof

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(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNeList(X)) → active(isNeList(X))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isList(X)) → active(isList(X))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(isQid(X)) → active(isQid(X))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(isNePal(X)) → active(isNePal(X))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isPalListKind(X)) → active(isPalListKind(X))
mark(isPal(X)) → active(isPal(X))
mark(a) → active(a)
mark(e) → active(e)
mark(i) → active(i)
mark(o) → active(o)
mark(u) → active(u)
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNeList(mark(X)) → isNeList(X)
isNeList(active(X)) → isNeList(X)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isList(mark(X)) → isList(X)
isList(active(X)) → isList(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
isQid(mark(X)) → isQid(X)
isQid(active(X)) → isQid(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isPalListKind(mark(X)) → isPalListKind(X)
isPalListKind(active(X)) → isPalListKind(X)
isPal(mark(X)) → isPal(X)
isPal(active(X)) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.