Term Rewriting System R:
[x]
f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(x)) -> G(f(x))
F(s(x)) -> F(x)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`

Dependency Pair:

F(s(x)) -> F(x)

Rules:

f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(s(x)) -> F(x)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(s(x1)) =  1 + x1 POL(F(x1)) =  x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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