Term Rewriting System R:
[x, y]
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(g(x)) -> F(f(x))
F(g(x)) -> F(x)
F'(s(x), y, y) -> F'(y, x, s(x))

Furthermore, R contains two SCCs.

R
DPs
→DP Problem 1
Argument Filtering and Ordering
→DP Problem 2
Remaining

Dependency Pairs:

F(g(x)) -> F(x)
F(g(x)) -> F(f(x))

Rules:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

The following dependency pairs can be strictly oriented:

F(g(x)) -> F(x)
F(g(x)) -> F(f(x))

The following usable rules w.r.t. to the AFS can be oriented:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> x1
h(x1) -> h

R
DPs
→DP Problem 1
AFS
→DP Problem 3
Dependency Graph
→DP Problem 2
Remaining

Dependency Pair:

Rules:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

Using the Dependency Graph resulted in no new DP problems.

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

F'(s(x), y, y) -> F'(y, x, s(x))

Rules:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

Termination of R could not be shown.
Duration:
0:00 minutes