Term Rewriting System R:
[x, y]
app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pair:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


Rules:


app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


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

app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rules:


app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes