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))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


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.


SCC1:

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))

On this Scc, 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 subcycle:


SCC1.Nar1:

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)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules 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(nil) = 0
POL(fcons) = 1
POL(fnil) = 0
POL(APP(x_1, x_2)) = 1 + x_1
POL(fmap) = 1
POL(app(x_1, x_2)) = x_1 + x_1*x_2
POL(cons) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 33.719 seconds.