Termination w.r.t. Q proof of /tmp/tmpcjOzpL/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__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)
ISPAL(V) → ACTIVATE(V)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
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)
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)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__isPal(X)) → ISPAL(X)
AND(tt, X) → ACTIVATE(X)
U71¹(tt, V) → ISNEPAL(activate(V))
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)

Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = x1
n____(x1, x2) = n____(x1, x2)
n__and(x1, x2) = n__and(x1, x2)
AND(x1, x2) = x2
activate(x1) = x1
ISNEPAL(x1) = ISNEPAL(x1)
U61¹(x1, x2) = U61¹(x1, x2)
isPalListKind(x1) = isPalListKind(x1)
ISPALLISTKIND(x1) = ISPALLISTKIND(x1)
n__isPal(x1) = n__isPal(x1)
ISPAL(x1) = ISPAL(x1)
tt = tt
n__isPalListKind(x1) = n__isPalListKind(x1)
isQid(x1) = x1
U71¹(x1, x2) = U71¹(x2)
and(x1, x2) = and(x1, x2)
n__nil = n__nil
nil = nil
u = u
n__u = n__u
__(x1, x2) = __(x1, x2)
e = e
n__e = n__e
a = a
n__a = n__a
o = o
n__o = n__o
i = i
n__i = n__i
isPal(x1) = isPal(x1)
U72(x1) = x1
U71(x1, x2) = U71(x2)
isNePal(x1) = isNePal(x1)
U62(x1) = U62
U61(x1, x2) = U61(x2)

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

[n_{_{_{_₂}}}, _{_₂}] > [n_{_and₂}, and₂]
[ISNEPAL₁, n_{_isPal₁}, ISPAL₁, U71^1₁, isPal₁, U71₁, isNePal₁, U61₁] > U61^1₂
[ISNEPAL₁, n_{_isPal₁}, ISPAL₁, U71^1₁, isPal₁, U71₁, isNePal₁, U61₁] > [isPalListKind₁, ISPALLISTKIND₁, n_{_{isPalListKind₁}}] > [n_{_and₂}, and₂]
[ISNEPAL₁, n_{_isPal₁}, ISPAL₁, U71^1₁, isPal₁, U71₁, isNePal₁, U61₁] > [isPalListKind₁, ISPALLISTKIND₁, n_{_{isPalListKind₁}}] > [tt, i, n_{_i}]
[ISNEPAL₁, n_{_isPal₁}, ISPAL₁, U71^1₁, isPal₁, U71₁, isNePal₁, U61₁] > U62 > [tt, i, n_{_i}]
[n_{_nil}, nil]
[u, n_{_u}] > [tt, i, n_{_i}]
[e, n_{_e}] > [tt, i, n_{_i}]
[a, n_{_a}] > [tt, i, n_{_i}]
[o, n_{_o}] > [tt, i, n_{_i}]

Status:

i: multiset
__₂: [1,2]
n_{_u}: multiset
n_{_i}: multiset
U61^1₂: multiset
n_{_nil}: multiset
and₂: multiset
n_{_a}: multiset
tt: multiset
ISPAL₁: [1]
isPalListKind₁: [1]
U71₁: [1]
nil: multiset
a: multiset
U62: []
U61₁: [1]
n_{_isPal₁}: [1]
e: multiset
n_{_e}: multiset
isNePal₁: [1]
o: multiset
n_{_{_{_₂}}}: [1,2]
n_{_and₂}: multiset
isPal₁: [1]
n_{_o}: multiset
u: multiset
ISPALLISTKIND₁: [1]
U71^1₁: [1]
n_{_{isPalListKind₁}}: [1]
ISNEPAL₁: [1]

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

                ↳ DependencyGraphProof

          ↳ QDP

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

U71¹(tt, V) → ISNEPAL(activate(V))
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
AND(tt, X) → ACTIVATE(X)
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.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 5 less nodes.

↳ 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))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U42¹(tt, V2) → ISNELIST(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))
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))
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))

Used ordering: Polynomial interpretation [25]:

POL(ISLIST(x₁)) = 1 + x₁
POL(ISNELIST(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U11¹(x₁, x₂)) = x₂
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U21¹(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U22(x₁, x₂)) = 0
POL(U22¹(x₁, x₂)) = 1 + x₂
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U41¹(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U42(x₁, x₂)) = 0
POL(U42¹(x₁, x₂)) = 1 + x₂
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U51¹(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U52(x₁, x₂)) = 0
POL(U52¹(x₁, x₂)) = 1 + x₂
POL(U53(x₁)) = 0
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(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(n____(x₁, x₂)) = 1 + x₁ + x₂
POL(n__a) = 0
POL(n__and(x₁, x₂)) = x₂
POL(n__e) = 0
POL(n__i) = 0
POL(n__isPal(x₁)) = 0
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:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
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
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))
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)
nil → n__nil
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:

U41¹(tt, V1, V2) → ISLIST(activate(V1))
U22¹(tt, V2) → ISLIST(activate(V2))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
U11¹(tt, V) → ISNELIST(activate(V))
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
U52¹(tt, V2) → ISLIST(activate(V2))
U41¹(tt, V1, V2) → U42¹(isList(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.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 10 less nodes.