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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(x, g(y)) -> F(g(x), y)

Furthermore, R contains one SCC.

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

Dependency Pair:

F(x, g(y)) -> F(g(x), y)

Rules:

f(x, a) -> x
f(x, g(y)) -> f(g(x), y)

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Instantiation Transformation`

Dependency Pair:

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

Rules:

f(x, a) -> x
f(x, g(y)) -> f(g(x), y)

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

f(x, a) -> x
f(x, g(y)) -> f(g(x), y)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(F(x1, x2)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

f(x, a) -> x
f(x, g(y)) -> f(g(x), y)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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