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

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

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

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

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

resulting in one new DP problem.
Used Argument Filtering System:
NOT(x1) -> NOT(x1)
or(x1, x2) -> or(x1, x2)
not(x1) -> not(x1)
and(x1, x2) -> and(x1, x2)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Dependency Graph

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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