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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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)
AND(x, or(y, z)) -> AND(x, y)
AND(x, or(y, z)) -> AND(x, z)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pairs:

AND(x, or(y, z)) -> AND(x, z)
AND(x, or(y, z)) -> AND(x, y)

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

AND(x, or(y, z)) -> AND(x, z)
AND(x, or(y, z)) -> AND(x, y)

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
AND(x1, x2) -> AND(x1, x2)
or(x1, x2) -> or(x1, x2)

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

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and 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:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

NOT(or(x, y)) -> NOT(y)
NOT(or(x, y)) -> NOT(x)
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: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
NOT(x1) -> NOT(x1)
or(x1, x2) -> or(x1, x2)
and(x1, x2) -> and(x1, x2)

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

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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