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))
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
↳Size-Change Principle
→DP Problem 2
↳SCP
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))
We number the DPs as follows:
- AND(x, or(y, z)) -> AND(x, z)
- AND(x, or(y, z)) -> AND(x, y)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
or(x1, x2) -> or(x1, x2)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳Size-Change Principle
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))
We number the DPs as follows:
- NOT(or(x, y)) -> NOT(y)
- NOT(or(x, y)) -> NOT(x)
- NOT(and(x, y)) -> NOT(y)
- NOT(and(x, y)) -> NOT(x)
and get the following Size-Change Graph(s): {4, 3, 2, 1} | , | {4, 3, 2, 1} |
---|
1 | > | 1 |
|
which lead(s) to this/these maximal multigraph(s): {4, 3, 2, 1} | , | {4, 3, 2, 1} |
---|
1 | > | 1 |
|
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
and(x1, x2) -> and(x1, x2)
or(x1, x2) -> or(x1, x2)
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes