Term Rewriting System R:
[f, x, y]
app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))

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(app(rec, f), x), app(s, y)) -> APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
APP(app(app(rec, f), x), app(s, y)) -> APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) -> APP(app(app(rec, f), x), y)

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(rec, f), x), app(s, y)) -> APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) -> APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) -> APP(app(f, app(s, y)), app(app(app(rec, f), x), y))


Rules:


app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(app(rec, f), x), app(s, y)) -> APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
three new Dependency Pairs are created:

APP(app(app(rec, app(app(rec, f''), x'')), x), app(s, y'')) -> APP(app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')), app(app(app(rec, app(app(rec, f''), x'')), x), y''))
APP(app(app(rec, f''), x''), app(s, 0)) -> APP(app(f'', app(s, 0)), x'')
APP(app(app(rec, f''), x''), app(s, app(s, y''))) -> APP(app(f'', app(s, app(s, y''))), app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')))

The transformation is resulting in one new DP problem: 



   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Nar
             ...
               →DP Problem 2
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(app(rec, f''), x''), app(s, app(s, y''))) -> APP(app(f'', app(s, app(s, y''))), app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')))
APP(app(app(rec, f''), x''), app(s, 0)) -> APP(app(f'', app(s, 0)), x'')
APP(app(app(rec, app(app(rec, f''), x'')), x), app(s, y'')) -> APP(app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')), app(app(app(rec, app(app(rec, f''), x'')), x), y''))
APP(app(app(rec, f), x), app(s, y)) -> APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) -> APP(app(app(rec, f), x), y)


Rules:


app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))


Strategy:

innermost



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