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. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳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. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)


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)


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




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Size-Change Principle


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)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
  2. APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
app(x1, x2) -> app(x1, x2)

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes