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
         ↳Size-Change Principle


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





We number the DPs as follows:
  1. F(g(x, y), z) -> F(y, z)
  2. F(g(x, y), z) -> F(x, z)
  3. F(f(x, y), z) -> F(y, z)
  4. F(f(x, y), z) -> F(x, f(y, z))
and get the following Size-Change Graph(s):
{4, 3, 2, 1} , {4, 3, 2, 1}
1>1
2=2
{4, 3, 2, 1} , {4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{4, 3, 2, 1} , {4, 3, 2, 1}
1>1
{4, 3, 2, 1} , {4, 3, 2, 1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
g(x1, x2) -> g(x1, x2)

We obtain no new DP problems.

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