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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x), y) -> F(f(x, y), y)
F(s(x), y) -> F(x, y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

F(s(x), y) -> F(x, y)
F(s(x), y) -> F(f(x, y), y)


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)


Strategy:

innermost




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

F(s(x), y) -> F(f(x, y), y)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pair:

F(s(x), y) -> F(x, y)


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(s(x), y) -> F(x, y)


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(s(x1))=  1 + x1  
  POL(F(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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