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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

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

The following dependency pairs can be strictly oriented:

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

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(c(x1, x2)) =  x1 + x2 POL(G(x1)) =  x1 POL(s(x1)) =  1 + x1 POL(F(x1)) =  x1

resulting in one new DP problem.

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

Dependency Pairs:

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

Rules:

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

Using the Dependency Graph the DP problem was split into 2 DP problems.

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

Dependency Pair:

G(c(x, s(y))) -> G(c(s(x), y))

Rules:

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

The following dependency pair can be strictly oriented:

G(c(x, s(y))) -> G(c(s(x), y))

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(c(x1, x2)) =  1 + x2 POL(G(x1)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

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(c(x1, x2)) =  x1 POL(s(x1)) =  1 + x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

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