Term Rewriting System R:
[f, x, h, t]
app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), 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(twice, f), x) -> APP(f, app(f, x))
APP(app(twice, f), x) -> APP(f, x)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
APP(app(map, f), app(app(cons, h), t)) -> APP(cons, app(f, h))
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(cons, app(f, x)), app(app(fmap, tf), x))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(cons, app(f, x))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(fmap, tf), x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(fmap, tf)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(twice, f), x) -> APP(f, x)
APP(app(twice, f), x) -> APP(f, app(f, x))


Rules:


app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
  2. APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
  3. APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
  4. APP(app(twice, f), x) -> APP(f, x)
  5. APP(app(twice, f), x) -> APP(f, app(f, x))
and get the following Size-Change Graph(s):
{5, 4, 1} , {5, 4, 1}
1>1
2=2
{2} , {2}
1=1
2>2
{3} , {3}
1>1
2>2
{5, 4, 1} , {5, 4, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2} , {2}
1=1
2>2
{5, 4, 1} , {5, 4, 1}
1>1
2=2
{5, 4, 1} , {5, 4, 1}
1>1
{3} , {3}
1>1
2>2
{2} , {3}
1>1
2>2
{3} , {2}
1>1
2>2
{5, 4, 1} , {3}
1>1
{5, 4, 1} , {3}
1>1
2>2
{3} , {5, 4, 1}
1>1
{3} , {5, 4, 1}
1>1
2>2
{2} , {2}
1>1
2>2
{5, 4, 1} , {5, 4, 1}
1>1
2>2
{2} , {5, 4, 1}
1>1
2>2
{2} , {5, 4, 1}
1>1
{3} , {2}
1>1
{3} , {3}
1>1
{2} , {2}
1>1
{2} , {3}
1>1

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
trivial

We obtain no new DP problems.

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