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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x)) -> F(x)
G(s(0)) -> G(f(s(0)))
G(s(0)) -> F(s(0))

Furthermore, R contains two SCCs. 


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


Dependency Pair:

F(s(x)) -> F(x)


Rules:


f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(s(x)) -> F(x)


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))=  x1  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Forward Instantiation Transformation


Dependency Pair:

G(s(0)) -> G(f(s(0)))


Rules:


f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))


Strategy:

innermost




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

G(s(0)) -> G(f(s(0)))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.


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