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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

G(a) -> G(b)
G(a) -> B
B -> F(a, a)
F(a, a) -> G(d)

Furthermore, R contains one SCC.

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

Dependency Pair:

G(a) -> G(b)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(a) -> G(b)

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  x1 POL(G(x1)) =  x1 POL(b) =  0 POL(d) =  0 POL(a) =  1 POL(f) =  0

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

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

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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