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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rule for innermost can be oriented:

f(f(x)) -> f(x)


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
0 > f

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
f(x1) -> f


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


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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