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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(g(x), g(y)) -> F(p(f(g(x), s(y))), g(s(p(x))))
F(g(x), g(y)) -> P(f(g(x), s(y)))
F(g(x), g(y)) -> F(g(x), s(y))
F(g(x), g(y)) -> G(s(p(x)))
F(g(x), g(y)) -> P(x)
P(0) -> G(0)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pair:

F(g(x), g(y)) -> F(p(f(g(x), s(y))), g(s(p(x))))

Rules:

f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
p(0) -> g(0)
g(s(p(x))) -> p(x)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(g(x), g(y)) -> F(p(f(g(x), s(y))), g(s(p(x))))
two new Dependency Pairs are created:

F(g(x''), g(y)) -> F(p(f(g(x''), s(y))), p(x''))
F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(s(g(0))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Narrowing Transformation`

Dependency Pairs:

F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(s(g(0))))
F(g(x''), g(y)) -> F(p(f(g(x''), s(y))), p(x''))

Rules:

f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
p(0) -> g(0)
g(s(p(x))) -> p(x)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(g(0), g(y)) -> F(p(f(g(0), s(y))), g(s(g(0))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

F(g(x''), g(y)) -> F(p(f(g(x''), s(y))), p(x''))

Rules:

f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
p(0) -> g(0)
g(s(p(x))) -> p(x)

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes