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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

A(x, g) -> A(f, a(g, a(f, x)))
two new Dependency Pairs are created:

A(a(f, x''), g) -> A(f, a(g, a(x'', g)))
A(g, g) -> A(f, a(g, a(f, a(g, a(f, f)))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

A(x, g) -> A(g, a(f, x))
two new Dependency Pairs are created:

A(a(f, x''), g) -> A(g, a(x'', g))
A(g, g) -> A(g, a(f, a(g, a(f, f))))

The transformation is resulting in two new DP problems:

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

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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) =  0 POL(a(x1, x2)) =  1 + x2 POL(A(x1, x2)) =  x1 + x2 POL(f) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Instantiation Transformation`

Dependency Pair:

A(x, g) -> A(f, x)

Rules:

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

Strategy:

innermost

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

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

A(f, g) -> A(f, f)

The transformation is resulting in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Narrowing Transformation`

Dependency Pair:

A(g, g) -> A(g, a(f, a(g, a(f, f))))

Rules:

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

Strategy:

innermost

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

A(g, g) -> A(g, a(f, a(g, a(f, f))))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

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