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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

A -> G(c)
F(g(X), b) -> F(a, X)
F(g(X), b) -> A

Furthermore, R contains one SCC.

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

Dependency Pair:

F(g(X), b) -> F(a, X)

Rules:

a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(g(X), b) -> F(a, X)

The following usable rules for innermost can be oriented:

a -> g(c)
g(a) -> b

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(c) =  0 POL(g(x1)) =  1 + x1 POL(b) =  1 POL(a) =  1 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(x1)
a -> a

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

Dependency Pair:

Rules:

a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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