Termination w.r.t. Q proof of /tmp/tmpfuVmkn/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:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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:

ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
U51¹(tt, V1, V2) → ACTIVATE(V2)
ISNELIST(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → ISQID(activate(I))
U51¹(tt, V1, V2) → ISNELIST(activate(V1))
U22¹(tt, V2) → U23¹(isList(activate(V2)))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U61¹(tt, V) → ISQID(activate(V))
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
ACTIVATE(n__o) → O
U61¹(tt, V) → U62¹(isQid(activate(V)))
U71¹(tt, V) → ISNEPAL(activate(V))
ISNEPAL(n____(I, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
U42¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
U61¹(tt, V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
__¹(__(X, Y), Z) → __¹(Y, Z)
U31¹(tt, V) → U32¹(isQid(activate(V)))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ACTIVATE(n__nil) → NIL
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPAL(V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)
U21¹(tt, V1, V2) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__i) → I
ACTIVATE(n__e) → E
U42¹(tt, V2) → ACTIVATE(V2)
U52¹(tt, V2) → ACTIVATE(V2)
U21¹(tt, V1, V2) → ACTIVATE(V2)
ISPAL(V) → ISPALLISTKIND(activate(V))
U42¹(tt, V2) → U43¹(isNeList(activate(V2)))
ISLIST(V) → ISPALLISTKIND(activate(V))
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
U31¹(tt, V) → ISQID(activate(V))
U11¹(tt, V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U22¹(tt, V2) → ISLIST(activate(V2))
U71¹(tt, V) → U72¹(isNePal(activate(V)))
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
U51¹(tt, V1, V2) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
ISNEPAL(V) → ACTIVATE(V)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
U31¹(tt, V) → ACTIVATE(V)
U41¹(tt, V1, V2) → ISLIST(activate(V1))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
ACTIVATE(n__a) → A
U11¹(tt, V) → U12¹(isNeList(activate(V)))
U41¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V2) → U53¹(isList(activate(V2)))
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V) → ACTIVATE(V)
U52¹(tt, V2) → ISLIST(activate(V2))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__u) → U
ACTIVATE(n____(X1, X2)) → __¹(activate(X1), activate(X2))
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
U41¹(tt, V1, V2) → ACTIVATE(V1)
ISNELIST(V) → ISPALLISTKIND(activate(V))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
U22¹(tt, V2) → ACTIVATE(V2)
ISLIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISNELIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
U71¹(tt, V) → ACTIVATE(V)
ISNELIST(V) → U31¹(isPalListKind(activate(V)), activate(V))
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
ISLIST(V) → ACTIVATE(V)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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:

ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
U51¹(tt, V1, V2) → ACTIVATE(V2)
ISNELIST(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → ISQID(activate(I))
U51¹(tt, V1, V2) → ISNELIST(activate(V1))
U22¹(tt, V2) → U23¹(isList(activate(V2)))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U61¹(tt, V) → ISQID(activate(V))
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
ACTIVATE(n__o) → O
U61¹(tt, V) → U62¹(isQid(activate(V)))
U71¹(tt, V) → ISNEPAL(activate(V))
ISNEPAL(n____(I, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
U42¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
U61¹(tt, V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
__¹(__(X, Y), Z) → __¹(Y, Z)
U31¹(tt, V) → U32¹(isQid(activate(V)))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ACTIVATE(n__nil) → NIL
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPAL(V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)
U21¹(tt, V1, V2) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__i) → I
ACTIVATE(n__e) → E
U42¹(tt, V2) → ACTIVATE(V2)
U52¹(tt, V2) → ACTIVATE(V2)
U21¹(tt, V1, V2) → ACTIVATE(V2)
ISPAL(V) → ISPALLISTKIND(activate(V))
U42¹(tt, V2) → U43¹(isNeList(activate(V2)))
ISLIST(V) → ISPALLISTKIND(activate(V))
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
U31¹(tt, V) → ISQID(activate(V))
U11¹(tt, V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U22¹(tt, V2) → ISLIST(activate(V2))
U71¹(tt, V) → U72¹(isNePal(activate(V)))
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
U51¹(tt, V1, V2) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
ISNEPAL(V) → ACTIVATE(V)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
U31¹(tt, V) → ACTIVATE(V)
U41¹(tt, V1, V2) → ISLIST(activate(V1))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
ACTIVATE(n__a) → A
U11¹(tt, V) → U12¹(isNeList(activate(V)))
U41¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V2) → U53¹(isList(activate(V2)))
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V) → ACTIVATE(V)
U52¹(tt, V2) → ISLIST(activate(V2))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__u) → U
ACTIVATE(n____(X1, X2)) → __¹(activate(X1), activate(X2))
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
U41¹(tt, V1, V2) → ACTIVATE(V1)
ISNELIST(V) → ISPALLISTKIND(activate(V))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
U22¹(tt, V2) → ACTIVATE(V2)
ISLIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISNELIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
U71¹(tt, V) → ACTIVATE(V)
ISNELIST(V) → U31¹(isPalListKind(activate(V)), activate(V))
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
ISLIST(V) → ACTIVATE(V)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 41 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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

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

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
__(X1, X2) → n____(X1, X2)

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:

__¹(__(X, Y), Z) → __¹(Y, Z)
The graph contains the following edges 1 > 1, 2 >= 2
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ACTIVATE(V)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNEPAL(V) → ACTIVATE(V)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
U71¹(tt, V) → ACTIVATE(V)
U71¹(tt, V) → ISNEPAL(activate(V))
ISNEPAL(n____(I, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
ISPAL(V) → ISPALLISTKIND(activate(V))
U61¹(tt, V) → ACTIVATE(V)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
ISNEPAL(n____(I, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)
ISNEPAL(V) → ACTIVATE(V)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISNEPAL(V) → ISPALLISTKIND(activate(V))
U71¹(tt, V) → ACTIVATE(V)
U71¹(tt, V) → ISNEPAL(activate(V))
ISPAL(V) → ISPALLISTKIND(activate(V))
U61¹(tt, V) → ACTIVATE(V)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( i ) =

/	0	\
\	0	/

M( n__u ) =

/	0	\
\	0	/

M( __(x₁, x₂) ) =

/	1	\
\	1	/

/	1	1	\
\	1	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__i ) =

/	0	\
\	0	/

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( U71(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( n__a ) =

/	1	\
\	0	/

M( tt ) =

/	0	\
\	0	/

M( isPalListKind(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U72(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( a ) =

/	1	\
\	0	/

M( n__isPal(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( e ) =

/	0	\
\	0	/

M( n__e ) =

/	0	\
\	0	/

M( isNePal(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	1	/

x₁

M( n____(x₁, x₂) ) =

/	1	\
\	1	/

/	1	1	\
\	1	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( o ) =

/	0	\
\	0	/

M( isQid(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

M( U62(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( isPal(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( n__o ) =

/	0	\
\	0	/

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	1	/

x₂

M( u ) =

/	0	\
\	0	/

M( n__isPalListKind(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

Tuple symbols:

M( ISPAL(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISPALLISTKIND(x₁) ) =

[

]

x₁

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( U71¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ACTIVATE(x₁) ) =

[

]

x₁

M( ISNEPAL(x₁) ) =

[

]

x₁

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

activate(n__nil) → nil
u → n__u
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
e → n__e
a → n__a
o → n__o
i → n__i
isPalListKind(X) → n__isPalListKind(X)
__(X1, X2) → n____(X1, X2)
isPal(X) → n__isPal(X)
and(X1, X2) → n__and(X1, X2)
isQid(n__o) → tt
isQid(n__i) → tt
nil → n__nil
isQid(n__u) → tt
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
U72(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U62(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isPalListKind(n__u) → tt
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

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

ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)
ISNEPAL(V) → ACTIVATE(V)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISNEPAL(V) → ISPALLISTKIND(activate(V))
U71¹(tt, V) → ACTIVATE(V)
U71¹(tt, V) → ISNEPAL(activate(V))
ISPAL(V) → ISPALLISTKIND(activate(V))
U61¹(tt, V) → ACTIVATE(V)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
ISPAL(V) → ACTIVATE(V)
ISNEPAL(V) → ACTIVATE(V)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
U71¹(tt, V) → ACTIVATE(V)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__isPal(X)) → ISPAL(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNEPAL(V) → ISPALLISTKIND(activate(V))
U71¹(tt, V) → ISNEPAL(activate(V))
ISPAL(V) → ISPALLISTKIND(activate(V))
U61¹(tt, V) → ACTIVATE(V)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNEPAL(x₁)) = 1 + x₁
POL(ISPAL(x₁)) = 1 + x₁
POL(ISPALLISTKIND(x₁)) = 1 + x₁
POL(U61(x₁, x₂)) = 0
POL(U61¹(x₁, x₂)) = x₂
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 1 + x₂
POL(U71¹(x₁, x₂)) = 1 + x₂
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 1
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 1
POL(i) = 1
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 1 + x₁
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = x₁
POL(n____(x₁, x₂)) = 1 + x₁ + x₂
POL(n__a) = 1
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__e) = 1
POL(n__i) = 1
POL(n__isPal(x₁)) = 1 + x₁
POL(n__isPalListKind(x₁)) = 0
POL(n__nil) = 0
POL(n__o) = 1
POL(n__u) = 0
POL(nil) = 0
POL(o) = 1
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

activate(n__nil) → nil
u → n__u
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
e → n__e
a → n__a
o → n__o
i → n__i
isPalListKind(X) → n__isPalListKind(X)
__(X1, X2) → n____(X1, X2)
isPal(X) → n__isPal(X)
and(X1, X2) → n__and(X1, X2)
isQid(n__o) → tt
isQid(n__i) → tt
nil → n__nil
isQid(n__u) → tt
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
U72(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U62(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isPalListKind(n__u) → tt
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

          ↳ QDP

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

U71¹(tt, V) → ISNEPAL(activate(V))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U61¹(tt, V) → ACTIVATE(V)
ISPAL(V) → ISPALLISTKIND(activate(V))
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNEPAL(V) → ISPALLISTKIND(activate(V))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

          ↳ QDP

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

ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

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

ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
The graph contains the following edges 1 > 2
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
The graph contains the following edges 1 > 1
AND(tt, X) → ACTIVATE(X)
The graph contains the following edges 2 >= 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

U22¹(tt, V2) → ISLIST(activate(V2))
U51¹(tt, V1, V2) → ISNELIST(activate(V1))
ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
U41¹(tt, V1, V2) → ISLIST(activate(V1))
U42¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
U11¹(tt, V) → ISNELIST(activate(V))
U52¹(tt, V2) → ISLIST(activate(V2))
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

U51¹(tt, V1, V2) → ISNELIST(activate(V1))
ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
U41¹(tt, V1, V2) → ISLIST(activate(V1))
U42¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))

The remaining pairs can at least be oriented weakly.

U22¹(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U11¹(tt, V) → ISNELIST(activate(V))
U52¹(tt, V2) → ISLIST(activate(V2))

Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2) = U22¹(x2)
tt = tt
ISLIST(x1) = ISLIST(x1)
activate(x1) = x1
U51¹(x1, x2, x3) = U51¹(x2, x3)
ISNELIST(x1) = ISNELIST(x1)
n____(x1, x2) = n____(x1, x2)
U41¹(x1, x2, x3) = U41¹(x2, x3)
and(x1, x2) = x2
isPalListKind(x1) = isPalListKind
n__isPalListKind(x1) = n__isPalListKind
U11¹(x1, x2) = U11¹(x2)
U52¹(x1, x2) = U52¹(x2)
isNeList(x1) = isNeList
U42¹(x1, x2) = U42¹(x1, x2)
isList(x1) = isList
U21¹(x1, x2, x3) = U21¹(x2, x3)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = U11
U12(x1) = U12
U21(x1, x2, x3) = U21
U22(x1, x2) = x1
U23(x1) = U23
U72(x1) = U72
U71(x1, x2) = U71(x1, x2)
isNePal(x1) = isNePal(x1)
U62(x1) = U62
U61(x1, x2) = U61
isQid(x1) = isQid
n__nil = n__nil
U42(x1, x2) = U42
U43(x1) = x1
U41(x1, x2, x3) = U41
U32(x1) = U32(x1)
U31(x1, x2) = U31(x1)
U53(x1) = x1
U52(x1, x2) = U52
U51(x1, x2, x3) = U51
n__o = n__o
n__e = n__e
n__i = n__i
n__a = n__a
n__u = n__u
isPal(x1) = isPal(x1)
n__and(x1, x2) = x2
n__isPal(x1) = n__isPal(x1)
u = u
e = e
a = a
o = o
i = i

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

[n_{_{_{_₂}}}, _{_₂}] > [U41^1₂, isList, U11, U21, U52] > [U22^1₁, tt, ISLIST₁, U51^1₂, ISNELIST₁, U11^1₁, U52^1₁, U42^1₂, U21^1₂, U12, U23, isQid] > U32₁
[isNeList, U42, U41] > [isPalListKind, n_{_{isPalListKind}}, isPal₁, n_{_isPal₁}] > [U72, U71₂] > [U22^1₁, tt, ISLIST₁, U51^1₂, ISNELIST₁, U11^1₁, U52^1₁, U42^1₂, U21^1₂, U12, U23, isQid] > U32₁
[isNeList, U42, U41] > U31₁ > U32₁
[isNeList, U42, U41] > U51 > [U41^1₂, isList, U11, U21, U52] > [U22^1₁, tt, ISLIST₁, U51^1₂, ISNELIST₁, U11^1₁, U52^1₁, U42^1₂, U21^1₂, U12, U23, isQid] > U32₁
[nil, n_{_nil}]
isNePal₁ > [isPalListKind, n_{_{isPalListKind}}, isPal₁, n_{_isPal₁}] > [U72, U71₂] > [U22^1₁, tt, ISLIST₁, U51^1₂, ISNELIST₁, U11^1₁, U52^1₁, U42^1₂, U21^1₂, U12, U23, isQid] > U32₁
isNePal₁ > [U62, U61] > [U22^1₁, tt, ISLIST₁, U51^1₂, ISNELIST₁, U11^1₁, U52^1₁, U42^1₂, U21^1₂, U12, U23, isQid] > U32₁
[n_{_o}, o]
[n_{_e}, e]
[n_{_i}, i] > [U22^1₁, tt, ISLIST₁, U51^1₂, ISNELIST₁, U11^1₁, U52^1₁, U42^1₂, U21^1₂, U12, U23, isQid] > U32₁
[n_{_a}, a]
[n_{_u}, u] > [U22^1₁, tt, ISLIST₁, U51^1₂, ISNELIST₁, U11^1₁, U52^1₁, U42^1₂, U21^1₂, U12, U23, isQid] > U32₁

Status:

i: multiset
U51^1₂: multiset
U11^1₁: multiset
__₂: [1,2]
n_{_u}: multiset
isPalListKind: []
isNeList: []
U42: []
n_{_i}: multiset
U21^1₂: multiset
U11: []
n_{_nil}: multiset
U71₂: multiset
n_{_a}: multiset
isList: []
U61: []
tt: multiset
U22^1₁: multiset
U72: multiset
U41: []
U23: multiset
isQid: multiset
nil: multiset
U21: []
U51: multiset
a: multiset
ISNELIST₁: multiset
U62: []
n_{_isPal₁}: [1]
e: multiset
U52: []
U12: multiset
n_{_e}: multiset
isNePal₁: [1]
n_{_{_{_₂}}}: [1,2]
o: multiset
U52^1₁: multiset
isPal₁: [1]
n_{_o}: multiset
U41^1₂: [2,1]
U42^1₂: multiset
U31₁: multiset
n_{_{isPalListKind}}: []
u: multiset
U32₁: [1]
ISLIST₁: multiset

The following usable rules [17] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U72(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U62(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isList(n__nil) → tt
isList(V) → U11(isPalListKind(activate(V)), activate(V))
U42(tt, V2) → U43(isNeList(activate(V2)))
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U32(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U53(tt) → tt
U52(tt, V2) → U53(isList(activate(V2)))
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U43(tt) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isPalListKind(n__u) → tt
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
activate(n__nil) → nil
u → n__u
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
e → n__e
a → n__a
o → n__o
i → n__i
isPalListKind(X) → n__isPalListKind(X)
__(X1, X2) → n____(X1, X2)
isPal(X) → n__isPal(X)
and(X1, X2) → n__and(X1, X2)
isQid(n__o) → tt
isQid(n__i) → tt
nil → n__nil
isQid(n__u) → tt
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

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

U22¹(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U11¹(tt, V) → ISNELIST(activate(V))
U52¹(tt, V2) → ISLIST(activate(V2))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 4 less nodes.