R
↳Dependency Pair Analysis
ATAIL(cons(X, XS)) -> MARK(XS)
MARK(zeros) -> AZEROS
MARK(tail(X)) -> ATAIL(mark(X))
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> ATAIL(mark(X))
ATAIL(cons(X, XS)) -> MARK(XS)
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
four new Dependency Pairs are created:
MARK(tail(X)) -> ATAIL(mark(X))
MARK(tail(zeros)) -> ATAIL(azeros)
MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(tail(0)) -> ATAIL(0)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Narrowing Transformation
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
ATAIL(cons(X, XS)) -> MARK(XS)
MARK(tail(zeros)) -> ATAIL(azeros)
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
two new Dependency Pairs are created:
MARK(tail(zeros)) -> ATAIL(azeros)
MARK(tail(zeros)) -> ATAIL(cons(0, zeros))
MARK(tail(zeros)) -> ATAIL(zeros)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 3
↳Narrowing Transformation
MARK(tail(zeros)) -> ATAIL(cons(0, zeros))
MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, XS)) -> MARK(XS)
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
five new Dependency Pairs are created:
MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
MARK(tail(tail(X'''))) -> ATAIL(tail(mark(X''')))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(0))) -> ATAIL(atail(0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 4
↳Argument Filtering and Ordering
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, XS)) -> MARK(XS)
MARK(tail(zeros)) -> ATAIL(cons(0, zeros))
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(cons(X1, X2)) -> MARK(X1)
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
azeros -> cons(0, zeros)
azeros -> zeros
POL(MARK(x1)) = 1 + x1 POL(0) = 0 POL(cons(x1, x2)) = 1 + x1 + x2 POL(a__zeros) = 1 POL(tail(x1)) = x1 POL(a__tail(x1)) = x1 POL(mark(x1)) = 1 + x1 POL(zeros) = 0 POL(A__TAIL(x1)) = x1
MARK(x1) -> MARK(x1)
ATAIL(x1) -> ATAIL(x1)
tail(x1) -> tail(x1)
atail(x1) -> atail(x1)
cons(x1, x2) -> cons(x1, x2)
mark(x1) -> mark(x1)
azeros -> azeros
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 5
↳Argument Filtering and Ordering
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, XS)) -> MARK(XS)
MARK(tail(zeros)) -> ATAIL(cons(0, zeros))
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(tail(X)) -> MARK(X)
MARK(tail(zeros)) -> ATAIL(cons(0, zeros))
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
azeros -> cons(0, zeros)
azeros -> zeros
POL(MARK(x1)) = x1 POL(0) = 0 POL(cons(x1, x2)) = x1 + x2 POL(a__zeros) = 0 POL(tail(x1)) = 1 + x1 POL(a__tail(x1)) = 1 + x1 POL(mark(x1)) = x1 POL(zeros) = 0 POL(A__TAIL(x1)) = x1
MARK(x1) -> MARK(x1)
ATAIL(x1) -> ATAIL(x1)
tail(x1) -> tail(x1)
atail(x1) -> atail(x1)
cons(x1, x2) -> cons(x1, x2)
mark(x1) -> mark(x1)
azeros -> azeros
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 6
↳Dependency Graph
ATAIL(cons(X, XS)) -> MARK(XS)
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0