(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) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, n____(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1)) = x1
POL(U21(x1, x2)) = 4 + x1 + x2
POL(U22(x1)) = x1
POL(U31(x1)) = 1 + x1
POL(U41(x1, x2)) = 4 + x1 + x2
POL(U42(x1)) = 1 + x1
POL(U51(x1, x2)) = 5 + x1 + x2
POL(U52(x1)) = 1 + x1
POL(U61(x1)) = 1 + x1
POL(U71(x1, x2)) = 6 + x1 + x2
POL(U72(x1)) = 1 + x1
POL(U81(x1)) = 1 + x1
POL(__(x1, x2)) = 6 + x1 + x2
POL(a) = 1
POL(activate(x1)) = x1
POL(e) = 1
POL(i) = 1
POL(isList(x1)) = 3 + x1
POL(isNeList(x1)) = 2 + x1
POL(isNePal(x1)) = 2 + x1
POL(isPal(x1)) = 4 + x1
POL(isQid(x1)) = x1
POL(n____(x1, x2)) = 6 + x1 + x2
POL(n__a) = 1
POL(n__e) = 1
POL(n__i) = 1
POL(n__nil) = 0
POL(n__o) = 1
POL(n__u) = 1
POL(nil) = 0
POL(o) = 1
POL(tt) = 0
POL(u) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
__(X, nil) → X
__(nil, X) → X
U21(tt, V2) → U22(isList(activate(V2)))
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, n____(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt) → tt
U22(tt) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
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__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1)) = 1 + x1
POL(U22(x1)) = 1 + x1
POL(__(x1, x2)) = x1 + x2
POL(a) = 0
POL(activate(x1)) = x1
POL(e) = 0
POL(i) = 0
POL(n____(x1, x2)) = x1 + x2
POL(n__a) = 0
POL(n__e) = 0
POL(n__i) = 0
POL(n__nil) = 0
POL(n__o) = 0
POL(n__u) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
U11(tt) → tt
U22(tt) → tt
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
__(__(X, Y), Z) → __(X, __(Y, Z))
nil → n__nil
__(X1, X2) → n____(X1, X2)
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__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(__(x1, x2)) = 2 + 2·x1 + x2
POL(a) = 1
POL(activate(x1)) = 1 + 2·x1
POL(e) = 1
POL(i) = 0
POL(n____(x1, x2)) = 2 + 2·x1 + x2
POL(n__a) = 0
POL(n__e) = 0
POL(n__i) = 0
POL(n__nil) = 1
POL(n__o) = 0
POL(n__u) = 2
POL(nil) = 2
POL(o) = 0
POL(u) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
__(__(X, Y), Z) → __(X, __(Y, Z))
nil → n__nil
a → n__a
e → n__e
activate(n__nil) → nil
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
__(X1, X2) → n____(X1, X2)
i → n__i
o → n__o
u → n__u
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__a) → a
activate(n__e) → e
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(__(x1, x2)) = x1 + x2
POL(a) = 0
POL(activate(x1)) = x1
POL(e) = 0
POL(i) = 1
POL(n____(x1, x2)) = x1 + x2
POL(n__a) = 1
POL(n__e) = 1
POL(n__i) = 0
POL(n__o) = 0
POL(n__u) = 0
POL(o) = 1
POL(u) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
i → n__i
o → n__o
u → n__u
activate(n__a) → a
activate(n__e) → e
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
__(X1, X2) → n____(X1, X2)
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
Q is empty.
(9) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(__(x1, x2)) = 1 + 2·x1 + 2·x2
POL(activate(x1)) = 2·x1
POL(n____(x1, x2)) = 1 + 2·x1 + 2·x2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
(10) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
__(X1, X2) → n____(X1, X2)
Q is empty.
(11) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(__(x1, x2)) = 1 + x1 + x2
POL(n____(x1, x2)) = x1 + x2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
__(X1, X2) → n____(X1, X2)
(12) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(13) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(14) TRUE