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`
`         ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`

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

The following dependency pairs can be strictly oriented:

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

Additionally, the following rules can be oriented:

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Polo`

Dependency Pair:

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

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polynomial Ordering`

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

The following dependency pairs can be strictly oriented:

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

Additionally, the following rules can be oriented:

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 4`
`             ↳Polynomial Ordering`

Dependency Pairs:

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

The following dependency pairs can be strictly oriented:

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

Additionally, the following rules can be oriented:

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 4`
`             ↳Polo`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`

Dependency Pair:

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

Using the Dependency Graph resulted in no new DP problems.

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