R
↳Dependency Pair Analysis
G(g(x)) -> G(h(g(x)))
G(g(x)) -> H(g(x))
H(h(x)) -> H(f(h(x), x))
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
G(g(x)) -> G(h(g(x)))
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x), x))
two new Dependency Pairs are created:
G(g(x)) -> G(h(g(x)))
G(g(h(g(x'')))) -> G(h(g(x'')))
G(g(g(x''))) -> G(h(g(h(g(x'')))))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Argument Filtering and Ordering
G(g(g(x''))) -> G(h(g(h(g(x'')))))
G(g(h(g(x'')))) -> G(h(g(x'')))
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x), x))
G(g(h(g(x'')))) -> G(h(g(x'')))
h(h(x)) -> h(f(h(x), x))
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
G(x1) -> G(x1)
g(x1) -> g(x1)
h(x1) -> x1
f(x1, x2) -> x1
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳AFS
...
→DP Problem 3
↳Remaining Obligation(s)
G(g(g(x''))) -> G(h(g(h(g(x'')))))
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x), x))