R
↳Dependency Pair Analysis
F(g(a)) -> F(s(g(b)))
F(g(a)) -> G(b)
G(x) -> F(g(x))
G(x) -> G(x)
R
↳DPs
→DP Problem 1
↳Forward Instantiation Transformation
G(x) -> G(x)
G(x) -> F(g(x))
F(g(a)) -> G(b)
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))
one new Dependency Pair is created:
G(x) -> F(g(x))
G(a) -> F(g(a))
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳Argument Filtering and Ordering
F(g(a)) -> G(b)
G(a) -> F(g(a))
G(x) -> G(x)
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))
G(a) -> F(g(a))
g(x) -> f(g(x))
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
POL(g(x1)) = x1 POL(G(x1)) = 1 + x1 POL(b) = 0 POL(s(x1)) = x1 POL(a) = 1 POL(F(x1)) = x1 POL(f(x1)) = x1
G(x1) -> G(x1)
F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> f(x1)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳AFS
...
→DP Problem 3
↳Dependency Graph
F(g(a)) -> G(b)
G(x) -> G(x)
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳AFS
...
→DP Problem 4
↳Remaining Obligation(s)
G(x) -> G(x)
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))