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

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(s(x), y, y) -> F(y, x, s(x))

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pair:

F(s(x), y, y) -> F(y, x, s(x))

Rules:

g(x, y) -> x
g(x, y) -> y
f(s(x), y, y) -> f(y, x, s(x))

The following dependency pair can be strictly oriented:

F(s(x), y, y) -> F(y, x, s(x))

Additionally, the following rules can be oriented:

g(x, y) -> x
g(x, y) -> y
f(s(x), y, y) -> f(y, x, s(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1, x2)) =  x1 + x2 POL(s(x1)) =  1 + x1 POL(f(x1, x2, x3)) =  0 POL(F(x1, x2, x3)) =  x1 + x2

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Dependency Graph

Dependency Pair:

Rules:

g(x, y) -> x
g(x, y) -> y
f(s(x), y, y) -> f(y, x, s(x))

Using the Dependency Graph resulted in no new DP problems.

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