Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)
Q is empty.
Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
MARK(half(X)) → A__HALF(mark(X))
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
A__SQR(s(X)) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
MARK(dbl(X)) → A__DBL(mark(X))
MARK(add(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → MARK(X2)
MARK(recip(X)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SQR(s(X)) → A__SQR(mark(X))
MARK(terms(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
A__ADD(0, X) → MARK(X)
MARK(terms(X)) → A__TERMS(mark(X))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__DBL(s(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(cons(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
A__HALF(s(s(X))) → A__HALF(mark(X))
MARK(sqr(X)) → MARK(X)
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(half(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(X)
A__HALF(dbl(X)) → MARK(X)
A__TERMS(N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__HALF(s(s(X))) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__DBL(s(X)) → A__DBL(mark(X))
A__TERMS(N) → A__SQR(mark(N))
A__SQR(s(X)) → A__DBL(mark(X))
The TRS R consists of the following rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
Q DP problem:
The TRS P consists of the following rules:
MARK(half(X)) → A__HALF(mark(X))
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
A__SQR(s(X)) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
MARK(dbl(X)) → A__DBL(mark(X))
MARK(add(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → MARK(X2)
MARK(recip(X)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SQR(s(X)) → A__SQR(mark(X))
MARK(terms(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
A__ADD(0, X) → MARK(X)
MARK(terms(X)) → A__TERMS(mark(X))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__DBL(s(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(cons(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
A__HALF(s(s(X))) → A__HALF(mark(X))
MARK(sqr(X)) → MARK(X)
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(half(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(X)
A__HALF(dbl(X)) → MARK(X)
A__TERMS(N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__HALF(s(s(X))) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__DBL(s(X)) → A__DBL(mark(X))
A__TERMS(N) → A__SQR(mark(N))
A__SQR(s(X)) → A__DBL(mark(X))
The TRS R consists of the following rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
A__SQR(s(X)) → MARK(X)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
MARK(half(X)) → A__HALF(mark(X))
MARK(sqr(X)) → A__SQR(mark(X))
MARK(dbl(X)) → A__DBL(mark(X))
MARK(add(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → MARK(X2)
MARK(recip(X)) → MARK(X)
MARK(terms(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(X1, X2)) → MARK(X1)
A__ADD(0, X) → MARK(X)
MARK(terms(X)) → A__TERMS(mark(X))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__DBL(s(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(cons(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
A__HALF(s(s(X))) → A__HALF(mark(X))
MARK(sqr(X)) → MARK(X)
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(half(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(X)
A__HALF(dbl(X)) → MARK(X)
A__TERMS(N) → MARK(N)
MARK(add(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__HALF(s(s(X))) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__TERMS(N) → A__SQR(mark(N))
A__DBL(s(X)) → A__DBL(mark(X))
A__SQR(s(X)) → A__DBL(mark(X))
The TRS R consists of the following rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.