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))

Innermost 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
         ↳Usable Rules (Innermost)
       →DP Problem 2
         ↳UsableRules


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))


Strategy:

innermost




As we are in the innermost case, we can delete all 3 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳Size-Change Principle
       →DP Problem 2
         ↳UsableRules


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




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

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
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
         ↳UsableRules
       →DP Problem 2
         ↳Usable Rules (Innermost)


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))


Strategy:

innermost




As we are in the innermost case, we can delete all 3 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 4
             ↳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)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. NOT(or(x, y)) -> NOT(y)
  2. NOT(or(x, y)) -> NOT(x)
  3. NOT(and(x, y)) -> NOT(y)
  4. NOT(and(x, y)) -> NOT(x)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
{3} , {3}
1>1
{4} , {4}
1>1

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

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