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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

S(f(x, y)) -> F(s(y), s(x))
S(f(x, y)) -> S(y)
S(f(x, y)) -> S(x)
S(g(x, y)) -> G(s(x), s(y))
S(g(x, y)) -> S(x)
S(g(x, y)) -> S(y)
F(g(x, y), g(u, v)) -> G(f(x, u), f(y, v))
F(g(x, y), g(u, v)) -> F(x, u)
F(g(x, y), g(u, v)) -> F(y, v)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pairs:

F(g(x, y), g(u, v)) -> F(y, v)
F(g(x, y), g(u, v)) -> F(x, u)

Rules:

s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a

• Dependency Pairs:

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

Rules:

s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pairs:

F(g(x, y), g(u, v)) -> F(y, v)
F(g(x, y), g(u, v)) -> F(x, u)

Rules:

s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a

• Dependency Pairs:

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

Rules:

s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a

Termination of R could not be shown.
Duration:
0:00 minutes