Termination Proof using AProVE

Term Rewriting System R:
[x, f, t]
app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t), x))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(cons, app(f, x))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(fmap, t)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t), x))

Rules:

app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))

Strategy:

innermost

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

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t), x))

four new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)

Rules:

app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))

Strategy:

innermost

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

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)

five new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(fmap, app(app(fcons, f''), t'')), x'')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, f''), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t''''))), x')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Polynomial Ordering

Dependency Pairs:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, f''), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(fmap, app(app(fcons, f''), t'')), x'')
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))

Rules:

app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))

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

app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(fcons) = 0

POL(cons) = 0

POL(fnil) = 0

POL(fmap) = 1

POL(nil) = 0

POL(app(x₁, x₂)) = x₁

POL(APP(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

Rules:

app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))

Strategy:

innermost

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

POL(fcons)	= 0
POL(cons)	= 0
POL(fnil)	= 0
POL(fmap)	= 1
POL(nil)	= 0
POL(app(x₁, x₂))	= x₁
POL(APP(x₁, x₂))	= x₁