R
↳Dependency Pair Analysis
2ND(cons(X, XS)) -> HEAD(activate(XS))
2ND(cons(X, XS)) -> ACTIVATE(XS)
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(ntake(X1, X2)) -> TAKE(X1, X2)
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳Nar
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(ntake(X1, X2)) -> TAKE(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(ntake(X1, X2)) -> TAKE(X1, X2)
TAKE(x1, x2) -> x2
cons(x1, x2) -> cons(x1, x2)
ACTIVATE(x1) -> x1
ntake(x1, x2) -> ntake(x1, x2)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳Nar
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Narrowing Transformation
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
three new Dependency Pairs are created:
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, nfrom(X''))) -> SEL(N, from(X''))
SEL(s(N), cons(X, ntake(X1', X2'))) -> SEL(N, take(X1', X2'))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Nar
→DP Problem 4
↳Narrowing Transformation
SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, ntake(X1', X2'))) -> SEL(N, take(X1', X2'))
SEL(s(N), cons(X, nfrom(X''))) -> SEL(N, from(X''))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
two new Dependency Pairs are created:
SEL(s(N), cons(X, nfrom(X''))) -> SEL(N, from(X''))
SEL(s(N), cons(X, nfrom(X'''))) -> SEL(N, cons(X''', nfrom(s(X'''))))
SEL(s(N), cons(X, nfrom(X'''))) -> SEL(N, nfrom(X'''))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 5
↳Narrowing Transformation
SEL(s(N), cons(X, nfrom(X'''))) -> SEL(N, cons(X''', nfrom(s(X'''))))
SEL(s(N), cons(X, ntake(X1', X2'))) -> SEL(N, take(X1', X2'))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
three new Dependency Pairs are created:
SEL(s(N), cons(X, ntake(X1', X2'))) -> SEL(N, take(X1', X2'))
SEL(s(N), cons(X, ntake(0, X2''))) -> SEL(N, nil)
SEL(s(N), cons(X, ntake(s(N''), cons(X'', XS')))) -> SEL(N, cons(X'', ntake(N'', activate(XS'))))
SEL(s(N), cons(X, ntake(X1'', X2''))) -> SEL(N, ntake(X1'', X2''))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 6
↳Remaining Obligation(s)
SEL(s(N), cons(X, ntake(s(N''), cons(X'', XS')))) -> SEL(N, cons(X'', ntake(N'', activate(XS'))))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, nfrom(X'''))) -> SEL(N, cons(X''', nfrom(s(X'''))))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X