Term Rewriting System R:
[f, h, t, c]
app(app(map1, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map1, f), t))
app(app(app(map2, f), c), app(app(cons, h), t)) -> app(app(cons, app(app(f, h), c)), app(app(app(map2, f), c), t))
app(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map3, f), g), c), t))

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(map1, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map1, f), t))
APP(app(map1, f), app(app(cons, h), t)) -> APP(cons, app(f, h))
APP(app(map1, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map1, f), app(app(cons, h), t)) -> APP(app(map1, f), t)
APP(app(app(map2, f), c), app(app(cons, h), t)) -> APP(app(cons, app(app(f, h), c)), app(app(app(map2, f), c), t))
APP(app(app(map2, f), c), app(app(cons, h), t)) -> APP(cons, app(app(f, h), c))
APP(app(app(map2, f), c), app(app(cons, h), t)) -> APP(app(f, h), c)
APP(app(app(map2, f), c), app(app(cons, h), t)) -> APP(f, h)
APP(app(app(map2, f), c), app(app(cons, h), t)) -> APP(app(app(map2, f), c), t)
APP(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> APP(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map3, f), g), c), t))
APP(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> APP(cons, app(app(app(f, g), h), c))
APP(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> APP(app(app(f, g), h), c)
APP(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> APP(app(f, g), h)
APP(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> APP(f, g)
APP(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> APP(app(app(app(map3, f), g), c), t)

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> APP(app(app(app(map3, f), g), c), t)
APP(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> APP(app(f, g), h)
APP(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> APP(app(app(f, g), h), c)
APP(app(app(map2, f), c), app(app(cons, h), t)) -> APP(app(app(map2, f), c), t)
APP(app(app(map2, f), c), app(app(cons, h), t)) -> APP(f, h)
APP(app(app(map2, f), c), app(app(cons, h), t)) -> APP(app(f, h), c)
APP(app(map1, f), app(app(cons, h), t)) -> APP(app(map1, f), t)
APP(app(map1, f), app(app(cons, h), t)) -> APP(f, h)


Rules:


app(app(map1, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map1, f), t))
app(app(app(map2, f), c), app(app(cons, h), t)) -> app(app(cons, app(app(f, h), c)), app(app(app(map2, f), c), t))
app(app(app(app(map3, f), g), c), app(app(cons, h), t)) -> app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map3, f), g), c), t))


Strategy:

innermost



The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes