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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(f, app(g, x)) -> APP(g, app(f, app(f, x)))
APP(f, app(g, x)) -> APP(f, app(f, x))
APP(f, app(g, x)) -> APP(f, x)
APP(f, app(h, x)) -> APP(h, app(g, x))
APP(f, app(h, x)) -> APP(g, x)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳A-Transformation`

Dependency Pairs:

APP(f, app(g, x)) -> APP(f, x)
APP(f, app(g, x)) -> APP(f, app(f, x))

Rules:

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

Strategy:

innermost

We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳ATrans`
`           →DP Problem 2`
`             ↳Negative Polynomial Order`

Dependency Pairs:

F(g(x)) -> F(x)
F(g(x)) -> F(f(x))

Rules:

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

Strategy:

innermost

The following Dependency Pairs can be strictly oriented using the given order.

F(g(x)) -> F(x)
F(g(x)) -> F(f(x))

Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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

Used ordering:
Polynomial Order with Interpretation:

POL( F(x1) ) = x1

POL( g(x1) ) = x1 + 1

POL( f(x1) ) = x1

POL( h(x1) ) = 0

This results in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳ATrans`
`           →DP Problem 2`
`             ↳Neg POLO`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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