Term Rewriting System R:
[x, y, z]
and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.

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

Dependency Pair:

OR(x, or(y, y)) -> OR(x, y)

Rules:

and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)

The following dependency pair can be strictly oriented:

OR(x, or(y, y)) -> OR(x, y)

Additionally, the following rules can be oriented:

and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(and(x1, x2)) =  x1 + x2 POL(false) =  0 POL(OR(x1, x2)) =  x2 POL(or(x1, x2)) =  1 + x1 POL(true) =  0

resulting in one new DP problem.

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

Dependency Pair:

Rules:

and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

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

Rules:

and(x, or(y, z)) -> or(and(x, y), and(x, z))
and(x, and(y, y)) -> and(x, y)
and(x, true) -> x
and(false, y) -> false
and(x, x) -> x
or(or(x, y), and(y, z)) -> or(x, y)
or(x, and(x, y)) -> x
or(true, y) -> true
or(x, false) -> x
or(x, x) -> x
or(x, or(y, y)) -> or(x, y)

Termination of R could not be shown.
Duration:
0:00 minutes