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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(compose, f), g), x) -> APP(f, app(g, x))
APP(app(app(compose, f), g), x) -> APP(g, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

APP(app(app(compose, f), g), x) -> APP(g, x)
APP(app(app(compose, f), g), x) -> APP(f, app(g, x))


Rule:


app(app(app(compose, f), g), x) -> app(f, app(g, x))


Strategy:

innermost




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

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

APP(app(app(compose, f), app(app(compose, f''), g'')), x'') -> APP(app(app(compose, f''), g''), x'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Polynomial Ordering


Dependency Pairs:

APP(app(app(compose, f), app(app(compose, f''), g'')), x'') -> APP(app(app(compose, f''), g''), x'')
APP(app(app(compose, f), g), x) -> APP(f, app(g, x))


Rule:


app(app(app(compose, f), g), x) -> app(f, app(g, x))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(app(compose, f), app(app(compose, f''), g'')), x'') -> APP(app(app(compose, f''), g''), x'')
APP(app(app(compose, f), g), x) -> APP(f, app(g, x))


Additionally, the following usable rule for innermost can be oriented:

app(app(app(compose, f), g), x) -> app(f, app(g, x))


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

resulting in one new DP problem. 



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


Dependency Pair:


Rule:


app(app(app(compose, f), g), x) -> app(f, app(g, x))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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