Term Rewriting System R:
[X]
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Innermost 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)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`

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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(g(X)) -> F(X)
F(g(X)) -> F(f(X))

The following usable rules for innermost can be oriented:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(h) =  0 POL(F(x1)) =  1 + x1 POL(f(x1)) =  x1

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes