Term Rewriting System R:
[m, n, r]
p(m, n, s(r)) -> p(m, r, n)
p(m, s(n), 0) -> p(0, n, m)
p(m, 0, 0) -> m

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

P(m, n, s(r)) -> P(m, r, n)
P(m, s(n), 0) -> P(0, n, m)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

P(m, s(n), 0) -> P(0, n, m)
P(m, n, s(r)) -> P(m, r, n)


Rules:


p(m, n, s(r)) -> p(m, r, n)
p(m, s(n), 0) -> p(0, n, m)
p(m, 0, 0) -> m


Strategy:

innermost




The following dependency pairs can be strictly oriented:

P(m, s(n), 0) -> P(0, n, m)
P(m, n, s(r)) -> P(m, r, n)


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(P(x1, x2, x3))=  x1 + x2 + x3  
  POL(0)=  0  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
P(x1, x2, x3) -> P(x1, x2, x3)
s(x1) -> s(x1)


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


Dependency Pair:


Rules:


p(m, n, s(r)) -> p(m, r, n)
p(m, s(n), 0) -> p(0, n, m)
p(m, 0, 0) -> m


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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