Term Rewriting System R:
[x, y]
minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MINUS(h(x)) -> MINUS(x)
MINUS(f(x, y)) -> MINUS(y)
MINUS(f(x, y)) -> MINUS(x)

Furthermore, R contains one SCC.

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

Dependency Pairs:

MINUS(f(x, y)) -> MINUS(x)
MINUS(f(x, y)) -> MINUS(y)
MINUS(h(x)) -> MINUS(x)

Rules:

minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))

The following dependency pairs can be strictly oriented:

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

Additionally, the following rules can be oriented:

minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(minus(x1)) =  x1 POL(MINUS(x1)) =  1 + x1 POL(h(x1)) =  x1 POL(f(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

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

Dependency Pair:

MINUS(h(x)) -> MINUS(x)

Rules:

minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))

The following dependency pair can be strictly oriented:

MINUS(h(x)) -> MINUS(x)

Additionally, the following rules can be oriented:

minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))

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

resulting in one new DP problem.

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

Dependency Pair:

Rules:

minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))

Using the Dependency Graph resulted in no new DP problems.

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