Term Rewriting System R:
[y, x, z]
f(0, y) -> y
f(x, 0) -> x
f(i(x), y) -> i(x)
f(f(x, y), z) -> f(x, f(y, z))
f(g(x, y), z) -> g(f(x, z), f(y, z))
f(1, g(x, y)) -> x
f(2, g(x, y)) -> y

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(x, y), z) -> F(x, f(y, z))
F(f(x, y), z) -> F(y, z)
F(g(x, y), z) -> F(x, z)
F(g(x, y), z) -> F(y, z)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

F(g(x, y), z) -> F(y, z)
F(g(x, y), z) -> F(x, z)
F(f(x, y), z) -> F(y, z)
F(f(x, y), z) -> F(x, f(y, z))


Rules:


f(0, y) -> y
f(x, 0) -> x
f(i(x), y) -> i(x)
f(f(x, y), z) -> f(x, f(y, z))
f(g(x, y), z) -> g(f(x, z), f(y, z))
f(1, g(x, y)) -> x
f(2, g(x, y)) -> y





The following dependency pairs can be strictly oriented:

F(g(x, y), z) -> F(y, z)
F(g(x, y), z) -> F(x, z)
F(f(x, y), z) -> F(y, z)
F(f(x, y), z) -> F(x, f(y, z))


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

f(0, y) -> y
f(x, 0) -> x
f(i(x), y) -> i(x)
f(f(x, y), z) -> f(x, f(y, z))
f(g(x, y), z) -> g(f(x, z), f(y, z))
f(1, g(x, y)) -> x
f(2, g(x, y)) -> y


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

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


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


Dependency Pair:


Rules:


f(0, y) -> y
f(x, 0) -> x
f(i(x), y) -> i(x)
f(f(x, y), z) -> f(x, f(y, z))
f(g(x, y), z) -> g(f(x, z), f(y, z))
f(1, g(x, y)) -> x
f(2, g(x, y)) -> y





Using the Dependency Graph resulted in no new DP problems.

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