Term Rewriting System R:
[X, Y]
f(g(X), Y) -> f(X, f(g(X), Y))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(g(X), Y) -> F(X, f(g(X), Y))
F(g(X), Y) -> F(g(X), Y)

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(g(X), Y) -> F(g(X), Y)
F(g(X), Y) -> F(X, f(g(X), Y))

Rule:

f(g(X), Y) -> f(X, f(g(X), Y))

Strategy:

innermost

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

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

F(g(X), Y) -> F(X, f(X, f(g(X), Y)))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

F(g(X), Y) -> F(X, f(X, f(g(X), Y)))
F(g(X), Y) -> F(g(X), Y)

Rule:

f(g(X), Y) -> f(X, f(g(X), Y))

Strategy:

innermost

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

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

F(g(X), Y) -> F(X, f(X, f(X, f(g(X), Y))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

F(g(X), Y) -> F(X, f(X, f(X, f(g(X), Y))))
F(g(X), Y) -> F(g(X), Y)

Rule:

f(g(X), Y) -> f(X, f(g(X), Y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(g(X), Y) -> F(X, f(X, f(X, f(g(X), Y))))

Additionally, the following usable rule for innermost can be oriented:

f(g(X), Y) -> f(X, f(g(X), Y))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(f(x1, x2)) =  0 POL(F(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 4`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

F(g(X), Y) -> F(g(X), Y)

Rule:

f(g(X), Y) -> f(X, f(g(X), Y))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes