Term Rewriting System R:
[x, y]
or(x, x) -> x
and(x, x) -> x
not(not(x)) -> x
not(and(x, y)) -> or(not(x), not(y))
not(or(x, y)) -> and(not(x), not(y))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

NOT(and(x, y)) -> OR(not(x), not(y))
NOT(and(x, y)) -> NOT(x)
NOT(and(x, y)) -> NOT(y)
NOT(or(x, y)) -> AND(not(x), not(y))
NOT(or(x, y)) -> NOT(x)
NOT(or(x, y)) -> NOT(y)

Furthermore, R contains one SCC.

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

Dependency Pairs:

NOT(or(x, y)) -> NOT(y)
NOT(or(x, y)) -> NOT(x)
NOT(and(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(x)

Rules:

or(x, x) -> x
and(x, x) -> x
not(not(x)) -> x
not(and(x, y)) -> or(not(x), not(y))
not(or(x, y)) -> and(not(x), not(y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

NOT(or(x, y)) -> NOT(y)
NOT(or(x, y)) -> NOT(x)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(and(x1, x2)) =  x1 + x2 POL(NOT(x1)) =  x1 POL(or(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

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

Dependency Pairs:

NOT(and(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(x)

Rules:

or(x, x) -> x
and(x, x) -> x
not(not(x)) -> x
not(and(x, y)) -> or(not(x), not(y))
not(or(x, y)) -> and(not(x), not(y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

NOT(and(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(x)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(and(x1, x2)) =  1 + x1 + x2 POL(NOT(x1)) =  x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

or(x, x) -> x
and(x, x) -> x
not(not(x)) -> x
not(and(x, y)) -> or(not(x), not(y))
not(or(x, y)) -> and(not(x), not(y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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