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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(iterate, f), x) -> APP(app(cons, x), app(app(iterate, f), app(f, x)))
APP(app(iterate, f), x) -> APP(cons, x)
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) -> APP(f, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Rewriting Transformation


Dependency Pairs:

APP(app(iterate, f), x) -> APP(f, x)
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) -> APP(app(cons, x), app(app(iterate, f), app(f, x)))


Rule:


app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))


Strategy:

innermost




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

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

APP(app(iterate, f), x) -> APP(app(cons, x), app(app(cons, app(f, x)), app(app(iterate, f), app(f, app(f, x)))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rewriting Transformation


Dependency Pairs:

APP(app(iterate, f), x) -> APP(app(cons, x), app(app(cons, app(f, x)), app(app(iterate, f), app(f, app(f, x)))))
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) -> APP(f, x)


Rule:


app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))


Strategy:

innermost




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

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

APP(app(iterate, f), x) -> APP(app(cons, x), app(app(cons, app(f, x)), app(app(cons, app(f, app(f, x))), app(app(iterate, f), app(f, app(f, app(f, x)))))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(iterate, f), x) -> APP(app(cons, x), app(app(cons, app(f, x)), app(app(cons, app(f, app(f, x))), app(app(iterate, f), app(f, app(f, app(f, x)))))))
APP(app(iterate, f), x) -> APP(f, x)
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))


Rule:


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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 4
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(iterate, app(iterate, f'')), x'') -> APP(app(iterate, f''), x'')
APP(app(iterate, f), x) -> APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) -> APP(app(cons, x), app(app(cons, app(f, x)), app(app(cons, app(f, app(f, x))), app(app(iterate, f), app(f, app(f, app(f, x)))))))


Rule:


app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(iterate, f), x) -> APP(app(cons, x), app(app(cons, app(f, x)), app(app(cons, app(f, app(f, x))), app(app(iterate, f), app(f, app(f, app(f, x)))))))


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

app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rule:


app(app(iterate, f), x) -> app(app(cons, x), app(app(iterate, f), app(f, x)))


Strategy:

innermost



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