Term Rewriting System R:
[]
f(0) -> cons(0)
f(s(0)) -> f(p(s(0)))
p(s(0)) -> 0

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(0)) -> F(p(s(0)))
F(s(0)) -> P(s(0))

Furthermore, R contains one SCC.

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

Dependency Pair:

F(s(0)) -> F(p(s(0)))

Rules:

f(0) -> cons(0)
f(s(0)) -> f(p(s(0)))
p(s(0)) -> 0

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(s(0)) -> F(p(s(0)))

The following usable rule for innermost can be oriented:

p(s(0)) -> 0

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{0, p}

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
s(x1) -> s(x1)
p(x1) -> p

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

Dependency Pair:

Rules:

f(0) -> cons(0)
f(s(0)) -> f(p(s(0)))
p(s(0)) -> 0

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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