Term Rewriting System R:
[f, x]
app(app(apply, f), x) -> app(f, 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(apply, f), x) -> APP(f, x)

Furthermore, R contains one SCC. 


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


Dependency Pair:

APP(app(apply, f), x) -> APP(f, x)


Rule:


app(app(apply, f), x) -> app(f, x)


Strategy:

innermost




As we are in the innermost case, we can delete all 1 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Argument Filtering and Ordering


Dependency Pair:

APP(app(apply, f), x) -> APP(f, x)


Rule:

none


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(apply, f), x) -> APP(f, x)


There are no usable rules w.r.t. the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)
app(x1, x2) -> app(x1, x2)


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rule:

none


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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