Term Rewriting System R:
[x, y, z]
app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)
APP(app(app(f, 0), 1), x) -> APP(app(f, app(s, x)), x)
APP(app(app(f, 0), 1), x) -> APP(f, app(s, x))
APP(app(app(f, 0), 1), x) -> APP(s, x)
APP(app(app(f, x), y), app(s, z)) -> APP(s, app(app(app(f, 0), 1), z))
APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)
APP(app(app(f, x), y), app(s, z)) -> APP(f, 0)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)
APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, 0), 1), x) -> APP(app(f, app(s, x)), x)
APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)


Rules:


app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)
APP(app(app(f, 0), 1), x) -> APP(app(f, app(s, x)), x)


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))


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

resulting in one new DP problem. 



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


Dependency Pairs:

APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)


Rules:


app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)


Rules:


app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))


Strategy:

innermost



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