R
↳Dependency Pair Analysis
-'(s(x), s(y)) -> -'(x, y)
F(s(x)) -> -'(s(x), g(f(x)))
F(s(x)) -> G(f(x))
F(s(x)) -> F(x)
G(s(x)) -> -'(s(x), f(g(x)))
G(s(x)) -> F(g(x))
G(s(x)) -> G(x)
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳Nar
-'(s(x), s(y)) -> -'(x, y)
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
-'(s(x), s(y)) -> -'(x, y)
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
POL(-'(x1, x2)) = x1 + x2 POL(0) = 0 POL(g(x1)) = 1 + x1 POL(s(x1)) = 1 + x1 POL(f(x1)) = x1
-'(x1, x2) -> -'(x1, x2)
s(x1) -> s(x1)
-(x1, x2) -> x1
f(x1) -> f(x1)
g(x1) -> g(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳Nar
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Narrowing Transformation
G(s(x)) -> G(x)
F(s(x)) -> F(x)
G(s(x)) -> F(g(x))
F(s(x)) -> G(f(x))
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
two new Dependency Pairs are created:
F(s(x)) -> G(f(x))
F(s(0)) -> G(0)
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Nar
→DP Problem 4
↳Narrowing Transformation
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
F(s(x)) -> F(x)
G(s(x)) -> F(g(x))
G(s(x)) -> G(x)
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
two new Dependency Pairs are created:
G(s(x)) -> F(g(x))
G(s(0)) -> F(s(0))
G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 5
↳Argument Filtering and Ordering
G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
F(s(x)) -> F(x)
G(s(0)) -> F(s(0))
G(s(x)) -> G(x)
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
G(s(s(x''))) -> F(-(s(x''), f(g(x''))))
F(s(x)) -> F(x)
G(s(x)) -> G(x)
F(s(s(x''))) -> G(-(s(x''), g(f(x''))))
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))
POL(0) = 0 POL(g(x1)) = 1 + x1 POL(G(x1)) = x1 POL(s(x1)) = 1 + x1 POL(F(x1)) = x1 POL(f(x1)) = x1
G(x1) -> G(x1)
s(x1) -> s(x1)
F(x1) -> F(x1)
-(x1, x2) -> x1
f(x1) -> f(x1)
g(x1) -> g(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 6
↳Dependency Graph
G(s(0)) -> F(s(0))
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
f(0) -> 0
f(s(x)) -> -(s(x), g(f(x)))
g(0) -> s(0)
g(s(x)) -> -(s(x), f(g(x)))