R
↳Dependency Pair Analysis
F(f(x)) -> G(f(x))
G(g(x)) -> F(x)
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
G(g(x)) -> F(x)
F(f(x)) -> G(f(x))
f(f(x)) -> g(f(x))
g(g(x)) -> f(x)
one new Dependency Pair is created:
F(f(x)) -> G(f(x))
F(f(f(x''))) -> G(g(f(x'')))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Argument Filtering and Ordering
F(f(f(x''))) -> G(g(f(x'')))
G(g(x)) -> F(x)
f(f(x)) -> g(f(x))
g(g(x)) -> f(x)
G(g(x)) -> F(x)
g(g(x)) -> f(x)
f(f(x)) -> g(f(x))
POL(g(x1)) = 1 + x1 POL(G(x1)) = x1 POL(F(x1)) = x1 POL(f(x1)) = 1 + x1
G(x1) -> G(x1)
F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> f(x1)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳AFS
...
→DP Problem 3
↳Dependency Graph
F(f(f(x''))) -> G(g(f(x'')))
f(f(x)) -> g(f(x))
g(g(x)) -> f(x)