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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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

APP(app(apply, f), x) -> APP(f, x)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Argument Filtering and Ordering


Dependency Pair:

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


Rule:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rule for innermost can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(apply)=  0  
  POL(APP(x1, x2))=  1 + x1 + x2  
  POL(app(x1, x2))=  1 + x1 + x2  

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


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳AFS
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rule:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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