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

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(f(x, y, z), u, f(x, y, v)) -> F(x, y, f(z, u, v))
F(f(x, y, z), u, f(x, y, v)) -> F(z, u, v)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

F(f(x, y, z), u, f(x, y, v)) -> F(z, u, v)
F(f(x, y, z), u, f(x, y, v)) -> F(x, y, f(z, u, v))

Rules:

f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
f(x, y, y) -> y
f(x, y, g(y)) -> x
f(x, x, y) -> x
f(g(x), x, y) -> y

The following dependency pairs can be strictly oriented:

F(f(x, y, z), u, f(x, y, v)) -> F(z, u, v)
F(f(x, y, z), u, f(x, y, v)) -> F(x, y, f(z, u, v))

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

f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
f(x, y, y) -> y
f(x, y, g(y)) -> x
f(x, x, y) -> x
f(g(x), x, y) -> y

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

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

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

Dependency Pair:

Rules:

f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
f(x, y, y) -> y
f(x, y, g(y)) -> x
f(x, x, y) -> x
f(g(x), x, y) -> y

Using the Dependency Graph resulted in no new DP problems.

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