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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(0, 1, x) -> F(h(x), h(x), x)
F(0, 1, x) -> H(x)

Furthermore, R contains one SCC.

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

Dependency Pair:

F(0, 1, x) -> F(h(x), h(x), x)

Rules:

f(0, 1, x) -> f(h(x), h(x), x)
h(0) -> 0
h(g(x, y)) -> y

Strategy:

innermost

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

F(0, 1, x) -> F(h(x), h(x), x)
four new Dependency Pairs are created:

F(0, 1, 0) -> F(0, h(0), 0)
F(0, 1, g(x'', y')) -> F(y', h(g(x'', y')), g(x'', y'))
F(0, 1, 0) -> F(h(0), 0, 0)
F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))

The transformation is resulting in two new DP problems:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Rewriting Transformation`
`           →DP Problem 3`
`             ↳Rw`

Dependency Pair:

F(0, 1, 0) -> F(0, h(0), 0)

Rules:

f(0, 1, x) -> f(h(x), h(x), x)
h(0) -> 0
h(g(x, y)) -> y

Strategy:

innermost

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

F(0, 1, 0) -> F(0, h(0), 0)
one new Dependency Pair is created:

F(0, 1, 0) -> F(0, 0, 0)

The transformation is resulting in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Rw`
`           →DP Problem 3`
`             ↳Rewriting Transformation`

Dependency Pairs:

F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))
F(0, 1, g(x'', y')) -> F(y', h(g(x'', y')), g(x'', y'))

Rules:

f(0, 1, x) -> f(h(x), h(x), x)
h(0) -> 0
h(g(x, y)) -> y

Strategy:

innermost

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

F(0, 1, g(x'', y')) -> F(y', h(g(x'', y')), g(x'', y'))
one new Dependency Pair is created:

F(0, 1, g(x'', y')) -> F(y', y', g(x'', y'))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Rw`
`           →DP Problem 3`
`             ↳Rw`
`             ...`
`               →DP Problem 4`
`                 ↳Rewriting Transformation`

Dependency Pair:

F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))

Rules:

f(0, 1, x) -> f(h(x), h(x), x)
h(0) -> 0
h(g(x, y)) -> y

Strategy:

innermost

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

F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))
one new Dependency Pair is created:

F(0, 1, g(x'', y')) -> F(y', y', g(x'', y'))

The transformation is resulting in no new DP problems.

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