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:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__isList(nil) → tt
a__isPal(nil) → tt
a__isQid(o) → tt
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 2·x1 + x2
POL(a) = 0
POL(a____(x1, x2)) = 2·x1 + x2
POL(a__and(x1, x2)) = x1 + x2
POL(a__isList(x1)) = x1
POL(a__isNeList(x1)) = x1
POL(a__isNePal(x1)) = x1
POL(a__isPal(x1)) = 2·x1
POL(a__isQid(x1)) = x1
POL(and(x1, x2)) = x1 + x2
POL(e) = 0
POL(i) = 0
POL(isList(x1)) = x1
POL(isNeList(x1)) = x1
POL(isNePal(x1)) = x1
POL(isPal(x1)) = 2·x1
POL(isQid(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 2
POL(o) = 1
POL(tt) = 0
POL(u) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
a__isQid(e) → tt
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 2·x1 + x2
POL(a) = 0
POL(a____(x1, x2)) = 2·x1 + x2
POL(a__and(x1, x2)) = 2·x1 + x2
POL(a__isList(x1)) = x1
POL(a__isNeList(x1)) = x1
POL(a__isNePal(x1)) = 2·x1
POL(a__isPal(x1)) = 2·x1
POL(a__isQid(x1)) = x1
POL(and(x1, x2)) = 2·x1 + x2
POL(e) = 1
POL(i) = 0
POL(isList(x1)) = x1
POL(isNeList(x1)) = x1
POL(isNePal(x1)) = 2·x1
POL(isPal(x1)) = 2·x1
POL(isQid(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(i) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(i) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
a__isQid(i) → tt
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 2·x1 + x2
POL(a) = 0
POL(a____(x1, x2)) = 2·x1 + x2
POL(a__and(x1, x2)) = x1 + x2
POL(a__isList(x1)) = 2·x1
POL(a__isNeList(x1)) = 2·x1
POL(a__isNePal(x1)) = x1
POL(a__isPal(x1)) = 2·x1
POL(a__isQid(x1)) = x1
POL(and(x1, x2)) = x1 + x2
POL(e) = 1
POL(i) = 1
POL(isList(x1)) = 2·x1
POL(isNeList(x1)) = 2·x1
POL(isNePal(x1)) = x1
POL(isPal(x1)) = 2·x1
POL(isQid(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
a__isQid(a) → tt
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 2·x1 + x2
POL(a) = 1
POL(a____(x1, x2)) = 2·x1 + x2
POL(a__and(x1, x2)) = x1 + x2
POL(a__isList(x1)) = x1
POL(a__isNeList(x1)) = x1
POL(a__isNePal(x1)) = 2·x1
POL(a__isPal(x1)) = 2·x1
POL(a__isQid(x1)) = x1
POL(and(x1, x2)) = x1 + x2
POL(e) = 0
POL(i) = 0
POL(isList(x1)) = x1
POL(isNeList(x1)) = x1
POL(isNePal(x1)) = 2·x1
POL(isPal(x1)) = 2·x1
POL(isQid(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(o) = 2
POL(tt) = 0
POL(u) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isQid(u) → tt
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 2 + 2·x1 + x2
POL(a) = 0
POL(a____(x1, x2)) = 2 + 2·x1 + x2
POL(a__and(x1, x2)) = 2·x1 + x2
POL(a__isList(x1)) = x1
POL(a__isNeList(x1)) = x1
POL(a__isNePal(x1)) = x1
POL(a__isPal(x1)) = 2·x1
POL(a__isQid(x1)) = x1
POL(and(x1, x2)) = 2·x1 + x2
POL(e) = 0
POL(i) = 0
POL(isList(x1)) = x1
POL(isNeList(x1)) = x1
POL(isNePal(x1)) = x1
POL(isPal(x1)) = 2·x1
POL(isQid(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 1
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
a__and(tt, X) → mark(X)
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 2·x1 + x2
POL(a) = 0
POL(a____(x1, x2)) = 2·x1 + x2
POL(a__and(x1, x2)) = 2·x1 + 2·x2
POL(a__isList(x1)) = x1
POL(a__isNeList(x1)) = x1
POL(a__isNePal(x1)) = x1
POL(a__isPal(x1)) = x1
POL(a__isQid(x1)) = x1
POL(and(x1, x2)) = 2·x1 + 2·x2
POL(e) = 0
POL(i) = 0
POL(isList(x1)) = x1
POL(isNeList(x1)) = x1
POL(isNePal(x1)) = x1
POL(isPal(x1)) = x1
POL(isQid(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 1
POL(o) = 0
POL(tt) = 2
POL(u) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a__isList(V) → a__isNeList(V)
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
a__isList(V) → a__isNeList(V)
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = x1 + x2
POL(a) = 2
POL(a____(x1, x2)) = x1 + x2
POL(a__and(x1, x2)) = x1 + x2
POL(a__isList(x1)) = 2 + 2·x1
POL(a__isNeList(x1)) = 2 + 2·x1
POL(a__isNePal(x1)) = 2 + x1
POL(a__isPal(x1)) = 2 + 2·x1
POL(a__isQid(x1)) = 1 + x1
POL(and(x1, x2)) = x1 + x2
POL(e) = 1
POL(i) = 2
POL(isList(x1)) = 2 + x1
POL(isNeList(x1)) = 2 + x1
POL(isNePal(x1)) = 1 + x1
POL(isPal(x1)) = 1 + x1
POL(isQid(x1)) = 1 + x1
POL(mark(x1)) = 2·x1
POL(nil) = 2
POL(o) = 0
POL(tt) = 1
POL(u) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a__isList(V) → a__isNeList(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
a__isList(V) → a__isNeList(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
a__isList(V) → a__isNeList(V)
a__isPal(V) → a__isNePal(V)
mark(isNePal(X)) → a__isNePal(X)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = x1 + 2·x2
POL(a____(x1, x2)) = x1 + 2·x2
POL(a__and(x1, x2)) = 2·x1 + x2
POL(a__isList(x1)) = 2 + 2·x1
POL(a__isNeList(x1)) = 1 + x1
POL(a__isNePal(x1)) = x1
POL(a__isPal(x1)) = 2 + 2·x1
POL(a__isQid(x1)) = 1 + 2·x1
POL(and(x1, x2)) = 2·x1 + x2
POL(isList(x1)) = 1 + x1
POL(isNeList(x1)) = x1
POL(isNePal(x1)) = 1 + 2·x1
POL(isPal(x1)) = 2 + 2·x1
POL(isQid(x1)) = x1
POL(mark(x1)) = x1
POL(o) = 0
POL(u) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
mark(isPal(X)) → a__isPal(X)
mark(u) → u
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 2·x1 + x2
POL(a____(x1, x2)) = 2·x1 + x2
POL(a__and(x1, x2)) = x1 + 2·x2
POL(a__isPal(x1)) = 2·x1
POL(and(x1, x2)) = x1 + x2
POL(isPal(x1)) = 2 + 2·x1
POL(mark(x1)) = 2·x1
POL(o) = 0
POL(u) = 2
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(o) → o
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(o) → o
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
mark(o) → o
a__and(X1, X2) → and(X1, X2)
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 2·x1 + x2
POL(a____(x1, x2)) = 2·x1 + x2
POL(a__and(x1, x2)) = 2 + 2·x1 + 2·x2
POL(and(x1, x2)) = 1 + 2·x1 + 2·x2
POL(mark(x1)) = 2·x1
POL(o) = 1
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a____(X1, X2) → __(X1, X2)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a____(X1, X2) → __(X1, X2)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
mark(and(X1, X2)) → a__and(mark(X1), X2)
a____(X1, X2) → __(X1, X2)
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 1 + 2·x1 + 2·x2
POL(a____(x1, x2)) = 2 + 2·x1 + 2·x2
POL(a__and(x1, x2)) = 1 + x1 + x2
POL(and(x1, x2)) = 2 + 2·x1 + 2·x2
POL(mark(x1)) = 2·x1
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
Used ordering:
Polynomial interpretation [25]:
POL(__(x1, x2)) = 2 + x1 + 2·x2
POL(a____(x1, x2)) = 1 + x1 + x2
POL(mark(x1)) = 2·x1
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RisEmptyProof
Q restricted rewrite system:
R is empty.
Q is empty.
The TRS R is empty. Hence, termination is trivially proven.