Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
AND(or(y, z), x) -> AND(x, z)
AND(or(y, z), x) -> AND(x, y)
AND(x, or(y, z)) -> AND(x, z)
AND(x, or(y, z)) -> AND(x, y)

Rules:

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

Strategy:
innermost
Dependency Pairs:
NOT(or(x, y)) -> NOT(y)
NOT(or(x, y)) -> NOT(x)

Rules:

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

Strategy:
innermost
Dependency Pairs:
NOT(and(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(x)

Rules:

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

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
AND(or(y, z), x) -> AND(x, z)
AND(or(y, z), x) -> AND(x, y)
AND(x, or(y, z)) -> AND(x, z)
AND(x, or(y, z)) -> AND(x, y)

Rules:

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

Strategy:
innermost
Dependency Pairs:
NOT(or(x, y)) -> NOT(y)
NOT(or(x, y)) -> NOT(x)

Rules:

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

Strategy:
innermost
Dependency Pairs:
NOT(and(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(x)

Rules:

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

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
AND(or(y, z), x) -> AND(x, z)
AND(or(y, z), x) -> AND(x, y)
AND(x, or(y, z)) -> AND(x, z)
AND(x, or(y, z)) -> AND(x, y)

Rules:

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

Strategy:
innermost
Dependency Pairs:
NOT(or(x, y)) -> NOT(y)
NOT(or(x, y)) -> NOT(x)

Rules:

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

Strategy:
innermost
Dependency Pairs:
NOT(and(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(x)

Rules:

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

Strategy:
innermost

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