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)

Innermost 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 one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`

Dependency Pairs:

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)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

AND(x, or(y, z)) -> AND(x, y)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Forward Instantiation Transformation`

Dependency Pairs:

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

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)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

AND(x, or(y, z)) -> AND(x, z)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

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

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)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

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)) =  1 + x2 POL(false) =  1 POL(or(x1, x2)) =  1 + x1 + x2 POL(true) =  0 POL(AND(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

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

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)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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