Term Rewriting System R:
[f, x]
app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(iterate, f), x) -> APP(app(cons, x), app(app(iterate, f), app(f, x)))
APP(app(iterate, f), x) -> APP(cons, x)
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) -> APP(f, x)

Furthermore, R contains one SCC.

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

Dependency Pairs:

APP(app(iterate, f), x) -> APP(f, x)
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) -> APP(app(cons, x), app(app(iterate, f), app(f, x)))

Rule:

app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))

Strategy:

innermost

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

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

APP(app(iterate, f), x) -> APP(app(cons, x), app(app(cons, app(f, x)), app(app(iterate, f), app(f, app(f, x)))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

APP(app(iterate, f), x) -> APP(app(cons, x), app(app(cons, app(f, x)), app(app(iterate, f), app(f, app(f, x)))))
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) -> APP(f, x)

Rule:

app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), 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(iterate, f), x) -> APP(app(cons, x), app(app(cons, app(f, x)), app(app(iterate, f), app(f, app(f, x)))))
four new Dependency Pairs are created:

APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(cons, x''), app(app(iterate, f''), app(f'', x'')))), app(app(iterate, app(iterate, f'')), app(app(iterate, f''), app(app(iterate, f''), x'')))))
APP(app(iterate, f''), x'') -> APP(app(cons, x''), app(app(cons, app(f'', x'')), app(app(cons, app(f'', app(f'', x''))), app(app(iterate, f''), app(f'', app(f'', app(f'', x'')))))))
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, app(iterate, f'')), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, f''), app(f'', app(app(iterate, f''), x'')))))))
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, app(iterate, f'')), app(app(iterate, f''), app(app(cons, x''), app(app(iterate, f''), app(f'', x'')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, app(iterate, f'')), app(app(iterate, f''), app(app(cons, x''), app(app(iterate, f''), app(f'', x'')))))))
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, app(iterate, f'')), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, f''), app(f'', app(app(iterate, f''), x'')))))))
APP(app(iterate, f''), x'') -> APP(app(cons, x''), app(app(cons, app(f'', x'')), app(app(cons, app(f'', app(f'', x''))), app(app(iterate, f''), app(f'', app(f'', app(f'', x'')))))))
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(cons, x''), app(app(iterate, f''), app(f'', x'')))), app(app(iterate, app(iterate, f'')), app(app(iterate, f''), app(app(iterate, f''), x'')))))
APP(app(iterate, f), x) -> APP(f, x)
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))

Rule:

app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), 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(iterate, f), x) -> APP(f, x)
three new Dependency Pairs are created:

APP(app(iterate, app(iterate, f'')), x'') -> APP(app(iterate, f''), x'')
APP(app(iterate, app(iterate, app(iterate, f''''))), x') -> APP(app(iterate, app(iterate, f'''')), x')
APP(app(iterate, app(iterate, f'''')), x') -> APP(app(iterate, f''''), x')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Polynomial Ordering`

Dependency Pairs:

APP(app(iterate, app(iterate, f'''')), x') -> APP(app(iterate, f''''), x')
APP(app(iterate, app(iterate, app(iterate, f''''))), x') -> APP(app(iterate, app(iterate, f'''')), x')
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(iterate, f''), x'')
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, app(iterate, f'')), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, f''), app(f'', app(app(iterate, f''), x'')))))))
APP(app(iterate, f''), x'') -> APP(app(cons, x''), app(app(cons, app(f'', x'')), app(app(cons, app(f'', app(f'', x''))), app(app(iterate, f''), app(f'', app(f'', app(f'', x'')))))))
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(cons, x''), app(app(iterate, f''), app(f'', x'')))), app(app(iterate, app(iterate, f'')), app(app(iterate, f''), app(app(iterate, f''), x'')))))
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, app(iterate, f'')), app(app(iterate, f''), app(app(cons, x''), app(app(iterate, f''), app(f'', x'')))))))

Rule:

app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, app(iterate, f'')), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, f''), app(f'', app(app(iterate, f''), x'')))))))
APP(app(iterate, f''), x'') -> APP(app(cons, x''), app(app(cons, app(f'', x'')), app(app(cons, app(f'', app(f'', x''))), app(app(iterate, f''), app(f'', app(f'', app(f'', x'')))))))
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(cons, x''), app(app(iterate, f''), app(f'', x'')))), app(app(iterate, app(iterate, f'')), app(app(iterate, f''), app(app(iterate, f''), x'')))))
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(cons, x''), app(app(cons, app(app(iterate, f''), x'')), app(app(iterate, app(iterate, f'')), app(app(iterate, f''), app(app(cons, x''), app(app(iterate, f''), app(f'', x'')))))))

Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(iterate) =  1 POL(cons) =  0 POL(app(x1, x2)) =  x1 POL(APP(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

APP(app(iterate, app(iterate, f'''')), x') -> APP(app(iterate, f''''), x')
APP(app(iterate, app(iterate, app(iterate, f''''))), x') -> APP(app(iterate, app(iterate, f'''')), x')
APP(app(iterate, app(iterate, f'')), x'') -> APP(app(iterate, f''), x'')
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))

Rule:

app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:02 minutes