R
↳Dependency Pair Analysis
AF(s(0)) -> AF(ap(s(0)))
AF(s(0)) -> AP(s(0))
MARK(f(X)) -> AF(mark(X))
MARK(f(X)) -> MARK(X)
MARK(p(X)) -> AP(mark(X))
MARK(p(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳AFS
AF(s(0)) -> AF(ap(s(0)))
af(0) -> cons(0, f(s(0)))
af(s(0)) -> af(ap(s(0)))
af(X) -> f(X)
ap(s(0)) -> 0
ap(X) -> p(X)
mark(f(X)) -> af(mark(X))
mark(p(X)) -> ap(mark(X))
mark(0) -> 0
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
AF(s(0)) -> AF(ap(s(0)))
ap(s(0)) -> 0
ap(X) -> p(X)
POL(a__p) = 0 POL(0) = 0 POL(A__F(x1)) = x1 POL(s(x1)) = 1 + x1 POL(p) = 0
AF(x1) -> AF(x1)
s(x1) -> s(x1)
ap(x1) -> ap
p(x1) -> p
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳AFS
af(0) -> cons(0, f(s(0)))
af(s(0)) -> af(ap(s(0)))
af(X) -> f(X)
ap(s(0)) -> 0
ap(X) -> p(X)
mark(f(X)) -> af(mark(X))
mark(p(X)) -> ap(mark(X))
mark(0) -> 0
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Argument Filtering and Ordering
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(p(X)) -> MARK(X)
MARK(f(X)) -> MARK(X)
af(0) -> cons(0, f(s(0)))
af(s(0)) -> af(ap(s(0)))
af(X) -> f(X)
ap(s(0)) -> 0
ap(X) -> p(X)
mark(f(X)) -> af(mark(X))
mark(p(X)) -> ap(mark(X))
mark(0) -> 0
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
MARK(s(X)) -> MARK(X)
POL(MARK(x1)) = x1 POL(cons(x1, x2)) = x1 + x2 POL(s(x1)) = 1 + x1 POL(f(x1)) = x1 POL(p(x1)) = x1
MARK(x1) -> MARK(x1)
s(x1) -> s(x1)
cons(x1, x2) -> cons(x1, x2)
p(x1) -> p(x1)
f(x1) -> f(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳Argument Filtering and Ordering
MARK(cons(X1, X2)) -> MARK(X1)
MARK(p(X)) -> MARK(X)
MARK(f(X)) -> MARK(X)
af(0) -> cons(0, f(s(0)))
af(s(0)) -> af(ap(s(0)))
af(X) -> f(X)
ap(s(0)) -> 0
ap(X) -> p(X)
mark(f(X)) -> af(mark(X))
mark(p(X)) -> ap(mark(X))
mark(0) -> 0
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
POL(MARK(x1)) = x1 POL(cons(x1, x2)) = 1 + x1 + x2 POL(f(x1)) = x1 POL(p(x1)) = x1
MARK(x1) -> MARK(x1)
cons(x1, x2) -> cons(x1, x2)
p(x1) -> p(x1)
f(x1) -> f(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳AFS
...
→DP Problem 5
↳Argument Filtering and Ordering
MARK(p(X)) -> MARK(X)
MARK(f(X)) -> MARK(X)
af(0) -> cons(0, f(s(0)))
af(s(0)) -> af(ap(s(0)))
af(X) -> f(X)
ap(s(0)) -> 0
ap(X) -> p(X)
mark(f(X)) -> af(mark(X))
mark(p(X)) -> ap(mark(X))
mark(0) -> 0
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
MARK(p(X)) -> MARK(X)
POL(MARK(x1)) = x1 POL(f(x1)) = x1 POL(p(x1)) = 1 + x1
MARK(x1) -> MARK(x1)
p(x1) -> p(x1)
f(x1) -> f(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳AFS
...
→DP Problem 6
↳Argument Filtering and Ordering
MARK(f(X)) -> MARK(X)
af(0) -> cons(0, f(s(0)))
af(s(0)) -> af(ap(s(0)))
af(X) -> f(X)
ap(s(0)) -> 0
ap(X) -> p(X)
mark(f(X)) -> af(mark(X))
mark(p(X)) -> ap(mark(X))
mark(0) -> 0
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
MARK(f(X)) -> MARK(X)
POL(MARK(x1)) = x1 POL(f(x1)) = 1 + x1
MARK(x1) -> MARK(x1)
f(x1) -> f(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳AFS
...
→DP Problem 7
↳Dependency Graph
af(0) -> cons(0, f(s(0)))
af(s(0)) -> af(ap(s(0)))
af(X) -> f(X)
ap(s(0)) -> 0
ap(X) -> p(X)
mark(f(X)) -> af(mark(X))
mark(p(X)) -> ap(mark(X))
mark(0) -> 0
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))