Term Rewriting System R:
[f, x]
app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(F, app(app(F, f), x)), x) -> APP(app(F, app(G, app(app(F, f), x))), app(f, x))
APP(app(F, app(app(F, f), x)), x) -> APP(F, app(G, app(app(F, f), x)))
APP(app(F, app(app(F, f), x)), x) -> APP(G, app(app(F, f), x))
APP(app(F, app(app(F, f), x)), x) -> APP(f, x)

Furthermore, R contains one SCC.

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

Dependency Pairs:

APP(app(F, app(app(F, f), x)), x) -> APP(f, x)
APP(app(F, app(app(F, f), x)), x) -> APP(app(F, app(G, app(app(F, f), x))), app(f, x))

Rule:

app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))

Strategy:

innermost

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

APP(app(F, app(app(F, f), x)), x) -> APP(app(F, app(G, app(app(F, f), x))), app(f, x))
one new Dependency Pair is created:

APP(app(F, app(app(F, app(F, app(app(F, f''), x''))), x'')), x'') -> APP(app(F, app(G, app(app(F, app(F, app(app(F, f''), x''))), x''))), app(app(F, app(G, app(app(F, f''), x''))), app(f'', x'')))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

APP(app(F, app(app(F, app(F, app(app(F, f''), x''))), x'')), x'') -> APP(app(F, app(G, app(app(F, app(F, app(app(F, f''), x''))), x''))), app(app(F, app(G, app(app(F, f''), x''))), app(f'', x'')))
APP(app(F, app(app(F, f), x)), x) -> APP(f, x)

Rule:

app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))

Strategy:

innermost

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

APP(app(F, app(app(F, f), x)), x) -> APP(f, x)
two new Dependency Pairs are created:

APP(app(F, app(app(F, app(F, app(app(F, f''), x'''))), x0)), x0) -> APP(app(F, app(app(F, f''), x''')), x0)
APP(app(F, app(app(F, app(F, app(app(F, app(F, app(app(F, f''''), x'''0))), x'''''))), x')), x') -> APP(app(F, app(app(F, app(F, app(app(F, f''''), x'''0))), x''''')), x')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

APP(app(F, app(app(F, app(F, app(app(F, app(F, app(app(F, f''''), x'''0))), x'''''))), x')), x') -> APP(app(F, app(app(F, app(F, app(app(F, f''''), x'''0))), x''''')), x')
APP(app(F, app(app(F, app(F, app(app(F, f''), x'''))), x0)), x0) -> APP(app(F, app(app(F, f''), x''')), x0)
APP(app(F, app(app(F, app(F, app(app(F, f''), x''))), x'')), x'') -> APP(app(F, app(G, app(app(F, app(F, app(app(F, f''), x''))), x''))), app(app(F, app(G, app(app(F, f''), x''))), app(f'', x'')))

Rule:

app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))

Strategy:

innermost

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