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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

G(f(x), y) -> H(x, y)
H(x, y) -> G(x, f(y))

Furthermore, R contains one SCC.

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

Dependency Pairs:

H(x, y) -> G(x, f(y))
G(f(x), y) -> H(x, y)

Rules:

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

Strategy:

innermost

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

G(f(x), y) -> H(x, y)
one new Dependency Pair is created:

G(f(x''), f(y'')) -> H(x'', f(y''))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

G(f(x''), f(y'')) -> H(x'', f(y''))
H(x, y) -> G(x, f(y))

Rules:

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

Strategy:

innermost

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

H(x, y) -> G(x, f(y))
one new Dependency Pair is created:

H(x', f(y'''')) -> G(x', f(f(y'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 3`
`                 ↳Instantiation Transformation`

Dependency Pairs:

H(x', f(y'''')) -> G(x', f(f(y'''')))
G(f(x''), f(y'')) -> H(x'', f(y''))

Rules:

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

Strategy:

innermost

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

G(f(x''), f(y'')) -> H(x'', f(y''))
one new Dependency Pair is created:

G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 4`
`                 ↳Instantiation Transformation`

Dependency Pairs:

G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))
H(x', f(y'''')) -> G(x', f(f(y'''')))

Rules:

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

Strategy:

innermost

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

H(x', f(y'''')) -> G(x', f(f(y'''')))
one new Dependency Pair is created:

H(x'', f(f(y''''''''))) -> G(x'', f(f(f(y''''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`

Dependency Pairs:

H(x'', f(f(y''''''''))) -> G(x'', f(f(f(y''''''''))))
G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(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, x2)) =  x1 POL(H(x1, x2)) =  x1 POL(f(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 6`
`                 ↳Dependency Graph`

Dependency Pair:

H(x'', f(f(y''''''''))) -> G(x'', f(f(f(y''''''''))))

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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