Term Rewriting System R:
[x, y, z]
g(x, y) -> x
g(x, y) -> y
f(0, 1, x) -> f(s(x), x, x)
f(x, y, s(z)) -> s(f(0, 1, z))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(0, 1, x) -> F(s(x), x, x)
F(x, y, s(z)) -> F(0, 1, z)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

F(x, y, s(z)) -> F(0, 1, z)
F(0, 1, x) -> F(s(x), x, x)


Rules:


g(x, y) -> x
g(x, y) -> y
f(0, 1, x) -> f(s(x), x, x)
f(x, y, s(z)) -> s(f(0, 1, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(x, y, s(z)) -> F(0, 1, z)


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

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


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


Dependency Pair:

F(0, 1, x) -> F(s(x), x, x)


Rules:


g(x, y) -> x
g(x, y) -> y
f(0, 1, x) -> f(s(x), x, x)
f(x, y, s(z)) -> s(f(0, 1, z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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