Term Rewriting System R:
[x, y, z]
app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, x), app(app(+, y), z)) -> APP(+, app(app(*, x), y))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))


Rule:


app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))


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

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


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

resulting in one new DP problem. 



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




The following remains to be proven:
Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)


Rule:


app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Strategy:

innermost



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