Termination Proof using AProVE

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.

     ↳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.

     ↳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:

     ↳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:

     ↳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:

     ↳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.

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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:
0:03 minutes

POL(iterate)	= 1
POL(cons)	= 0
POL(app(x₁, x₂))	= x₁
POL(APP(x₁, x₂))	= x₁