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
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS


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 AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS


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
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering


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




The following dependency pair can be strictly oriented:

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


The following usable rules for innermost w.r.t. to the AFS can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{f, 0}

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 4
             ↳Dependency Graph


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.

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