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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

APP(f, app(g, x)) -> APP(f, app(f, x))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

APP(f, app(g, app(h, x''))) -> APP(f, app(h, app(g, x'')))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


app(f, app(g, x)) -> app(g, app(f, app(f, x)))
app(f, app(h, x)) -> app(h, 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)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

APP(f, app(g, app(g, x''))) -> APP(f, app(g, app(f, app(f, x''))))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Rewriting Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Rewriting Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 7
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

APP(f, app(g, app(g, app(h, x')))) -> APP(f, app(g, app(h, app(g, app(g, x')))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes