Term Rewriting System R:
[f, x, xs, g]
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)

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(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
APP(app(app(comp, f), g), x) -> APP(g, x)
APP(twice, f) -> APP(app(comp, f), f)
APP(twice, f) -> APP(comp, f)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

APP(app(app(comp, f), g), x) -> APP(g, x)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(app(comp, f), g), x) -> APP(g, x)
  2. APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
  3. APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
  4. APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1
2=2
{2, 1} , {2, 1}
1>1
{3} , {3}
1=1
2>2
{4} , {4}
1>1
2>2

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