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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Instantiation Transformation`

Dependency Pairs:

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

Rules:

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

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

G(s(x), y) -> G(f(x, y), 0)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

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

Rules:

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

The following dependency pairs can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pairs:

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

Rules:

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

The following dependency pairs can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

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

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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