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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(f(x)) -> F(g(f(x), x))
F(f(x)) -> G(f(x), x)
F(f(x)) -> F(h(f(x), f(x)))
F(f(x)) -> H(f(x), f(x))
H(x, x) -> G(x, 0)

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(f(x)) -> F(h(f(x), f(x)))
F(f(x)) -> F(g(f(x), x))

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Strategy:

innermost

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

F(f(x)) -> F(g(f(x), x))
one new Dependency Pair is created:

F(f(x'')) -> F(x'')

The transformation is resulting in one new DP problem:

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

Dependency Pair:

F(f(x)) -> F(h(f(x), f(x)))

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Strategy:

innermost

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

F(f(x)) -> F(h(f(x), f(x)))
one new Dependency Pair is created:

F(f(x'')) -> F(g(f(x''), 0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Narrowing Transformation`

Dependency Pair:

F(f(x'')) -> F(g(f(x''), 0))

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Strategy:

innermost

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

F(f(x'')) -> F(g(f(x''), 0))
one new Dependency Pair is created:

F(f(x''')) -> F(0)

The transformation is resulting in no new DP problems.

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