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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

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


Rules:


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

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rules for innermost can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
f > g

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:


Rules:


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