(0) Obligation:

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, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(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
niln__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
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.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

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, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(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
niln__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
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.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 52 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(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
niln__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
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.

(6) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
__(x1, x2)  =  __(x1, x2)

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

  • __1(__(X, Y), Z) → __1(Y, Z) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 > 1, 2 >= 2

  • __1(__(X, Y), Z) → __1(X, __(Y, Z)) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

We oriented the following set of usable rules [AAECC05,FROCOS05]. none

(7) TRUE

(8) Obligation:

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

ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → 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, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(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
niln__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
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.

(9) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Polynomial interpretation [POLO]:


POL(__(x1, x2)) = 1 + x1 + x2   
POL(a) = 1   
POL(activate(x1)) = x1   
POL(and(x1, x2)) = 1 + x2   
POL(e) = 1   
POL(i) = 1   
POL(isPalListKind(x1)) = x1   
POL(n____(x1, x2)) = 1 + x1 + x2   
POL(n__a) = 1   
POL(n__and(x1, x2)) = 1 + x2   
POL(n__e) = 1   
POL(n__i) = 1   
POL(n__isPalListKind(x1)) = x1   
POL(n__nil) = 1   
POL(n__o) = 1   
POL(n__u) = 1   
POL(nil) = 1   
POL(o) = 1   
POL(tt) = 1   
POL(u) = 1   

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

  • AND(tt, X) → ACTIVATE(X) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1)) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

  • ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))) (allowed arguments on rhs = {2})
    The graph contains the following edges 1 > 2

  • ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 >= 1

  • ACTIVATE(n__and(X1, X2)) → AND(X1, X2) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 > 2

  • ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

  • ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

We oriented the following set of usable rules [AAECC05,FROCOS05].


un__u
on__o
niln__nil
isPalListKind(X) → n__isPalListKind(X)
isPalListKind(n__u) → tt
isPalListKind(n__o) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__i) → tt
isPalListKind(n__e) → tt
isPalListKind(n__a) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
in__i
en__e
and(X1, X2) → n__and(X1, X2)
and(tt, X) → activate(X)
activate(X) → X
activate(n__u) → u
activate(n__o) → o
activate(n__nil) → nil
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__i) → i
activate(n__e) → e
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n____(X1, X2)) → __(X1, X2)
an__a
__(X1, X2) → n____(X1, X2)
__(X, nil) → X
__(nil, X) → X
__(__(X, Y), Z) → __(X, __(Y, Z))

(10) TRUE

(11) Obligation:

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

ISNEPAL(n____(I, __(P, I))) → ISPAL(activate(P))
ISPAL(V) → U711(isPalListKind(activate(V)), activate(V))
U711(tt, V) → ISNEPAL(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, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(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
niln__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
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.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Polynomial interpretation [POLO]:


POL(__(x1, x2)) = 1 + x1 + x2   
POL(a) = 1   
POL(activate(x1)) = x1   
POL(and(x1, x2)) = x2   
POL(e) = 1   
POL(i) = 1   
POL(isPalListKind(x1)) = x1   
POL(n____(x1, x2)) = 1 + x1 + x2   
POL(n__a) = 1   
POL(n__and(x1, x2)) = x2   
POL(n__e) = 1   
POL(n__i) = 1   
POL(n__isPalListKind(x1)) = x1   
POL(n__nil) = 1   
POL(n__o) = 1   
POL(n__u) = 1   
POL(nil) = 1   
POL(o) = 1   
POL(tt) = 1   
POL(u) = 1   

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

  • ISPAL(V) → U711(isPalListKind(activate(V)), activate(V)) (allowed arguments on rhs = {2})
    The graph contains the following edges 1 >= 2

  • U711(tt, V) → ISNEPAL(activate(V)) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • ISNEPAL(n____(I, __(P, I))) → ISPAL(activate(P)) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

We oriented the following set of usable rules [AAECC05,FROCOS05].


un__u
on__o
niln__nil
isPalListKind(X) → n__isPalListKind(X)
isPalListKind(n__u) → tt
isPalListKind(n__o) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__i) → tt
isPalListKind(n__e) → tt
isPalListKind(n__a) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
in__i
en__e
and(X1, X2) → n__and(X1, X2)
and(tt, X) → activate(X)
activate(X) → X
activate(n__u) → u
activate(n__o) → o
activate(n__nil) → nil
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__i) → i
activate(n__e) → e
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n____(X1, X2)) → __(X1, X2)
an__a
__(X1, X2) → n____(X1, X2)
__(X, nil) → X
__(nil, X) → X
__(__(X, Y), Z) → __(X, __(Y, Z))

(13) TRUE

(14) Obligation:

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

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

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, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(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
niln__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
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.

(15) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Polynomial interpretation [POLO]:


POL(__(x1, x2)) = 1 + x1 + x2   
POL(a) = 1   
POL(activate(x1)) = x1   
POL(and(x1, x2)) = x2   
POL(e) = 1   
POL(i) = 1   
POL(isPalListKind(x1)) = x1   
POL(n____(x1, x2)) = 1 + x1 + x2   
POL(n__a) = 1   
POL(n__and(x1, x2)) = x2   
POL(n__e) = 1   
POL(n__i) = 1   
POL(n__isPalListKind(x1)) = x1   
POL(n__nil) = 1   
POL(n__o) = 1   
POL(n__u) = 1   
POL(nil) = 1   
POL(o) = 1   
POL(tt) = 1   
POL(u) = 1   

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

  • ISLIST(V) → U111(isPalListKind(activate(V)), activate(V)) (allowed arguments on rhs = {2})
    The graph contains the following edges 1 >= 2

  • ISNELIST(n____(V1, V2)) → U411(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) (allowed arguments on rhs = {2, 3})
    The graph contains the following edges 1 > 2, 1 > 3

  • ISNELIST(n____(V1, V2)) → U511(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) (allowed arguments on rhs = {2, 3})
    The graph contains the following edges 1 > 2, 1 > 3

  • U421(tt, V2) → ISNELIST(activate(V2)) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • U411(tt, V1, V2) → U421(isList(activate(V1)), activate(V2)) (allowed arguments on rhs = {2})
    The graph contains the following edges 3 >= 2

  • U411(tt, V1, V2) → ISLIST(activate(V1)) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • U511(tt, V1, V2) → ISNELIST(activate(V1)) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • U111(tt, V) → ISNELIST(activate(V)) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • U511(tt, V1, V2) → U521(isNeList(activate(V1)), activate(V2)) (allowed arguments on rhs = {2})
    The graph contains the following edges 3 >= 2

  • U521(tt, V2) → ISLIST(activate(V2)) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • ISLIST(n____(V1, V2)) → U211(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) (allowed arguments on rhs = {2, 3})
    The graph contains the following edges 1 > 2, 1 > 3

  • U211(tt, V1, V2) → U221(isList(activate(V1)), activate(V2)) (allowed arguments on rhs = {2})
    The graph contains the following edges 3 >= 2

  • U211(tt, V1, V2) → ISLIST(activate(V1)) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • U221(tt, V2) → ISLIST(activate(V2)) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

We oriented the following set of usable rules [AAECC05,FROCOS05].


un__u
on__o
niln__nil
isPalListKind(X) → n__isPalListKind(X)
isPalListKind(n__u) → tt
isPalListKind(n__o) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__i) → tt
isPalListKind(n__e) → tt
isPalListKind(n__a) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
in__i
en__e
and(X1, X2) → n__and(X1, X2)
and(tt, X) → activate(X)
activate(X) → X
activate(n__u) → u
activate(n__o) → o
activate(n__nil) → nil
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__i) → i
activate(n__e) → e
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n____(X1, X2)) → __(X1, X2)
an__a
__(X1, X2) → n____(X1, X2)
__(X, nil) → X
__(nil, X) → X
__(__(X, Y), Z) → __(X, __(Y, Z))

(16) TRUE