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))

Termination of R to be shown. 



   R
     ↳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 two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP


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))





We number the DPs as follows:
  1. AND(or(y, z), x) -> AND(x, z)
  2. AND(or(y, z), x) -> AND(x, y)
  3. AND(x, or(y, z)) -> AND(x, z)
  4. AND(x, or(y, z)) -> AND(x, y)
and get the following Size-Change Graph(s):
{4, 3, 2, 1} , {4, 3, 2, 1}
1>2
2=1
{4, 3, 2, 1} , {4, 3, 2, 1}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{4, 3, 2, 1} , {4, 3, 2, 1}
1=1
2>2
{4, 3, 2, 1} , {4, 3, 2, 1}
1>1
2>2

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





We number the DPs as follows:
  1. NOT(and(x, y)) -> NOT(y)
  2. NOT(and(x, y)) -> NOT(x)
  3. NOT(or(x, y)) -> NOT(y)
  4. NOT(or(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