Term Rewriting System R:
[x, y]
f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(x), s(y)) -> F(x, y)
G(0, x) -> G(f(x, x), x)
G(0, x) -> F(x, x)

Furthermore, R contains two SCCs.

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

Dependency Pair:

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

Rules:

f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)

The following dependency pair can be strictly oriented:

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

Additionally, the following rules can be oriented:

f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)

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

f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

G(0, x) -> G(f(x, x), x)

Rules:

f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)

The following dependency pair can be strictly oriented:

G(0, x) -> G(f(x, x), x)

Additionally, the following rules can be oriented:

f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)

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

resulting in one new DP problem.

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

Dependency Pair:

Rules:

f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)

Using the Dependency Graph resulted in no new DP problems.

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