Term Rewriting System R:
[y]
f(a, y) -> f(y, g(y))
g(a) -> b
g(b) -> b

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(a, y) -> F(y, g(y))
F(a, y) -> G(y)

Furthermore, R contains one SCC.

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

Dependency Pair:

F(a, y) -> F(y, g(y))

Rules:

f(a, y) -> f(y, g(y))
g(a) -> b
g(b) -> b

The following dependency pair can be strictly oriented:

F(a, y) -> F(y, g(y))

The following rules can be oriented:

g(a) -> b
g(b) -> b
f(a, y) -> f(y, g(y))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g) =  0 POL(b) =  0 POL(a) =  1 POL(F(x1, x2)) =  x1 + x2 POL(f(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
g(x1) -> g
f(x1, x2) -> f(x1, x2)

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

Dependency Pair:

Rules:

f(a, y) -> f(y, g(y))
g(a) -> b
g(b) -> b

Using the Dependency Graph resulted in no new DP problems.

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