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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


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


Dependency Pairs:

APP(app(and, app(app(or, y), z)), x) -> APP(app(and, x), z)
APP(app(and, app(app(or, y), z)), x) -> APP(app(and, x), y)
APP(app(and, x), app(app(or, y), z)) -> APP(app(and, x), z)
APP(app(and, x), app(app(or, y), z)) -> APP(app(and, x), y)


Rules:


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





The original DP problem is in applicative form. Its DPs and usable rules are the following.

APP(app(and, app(app(or, y), z)), x) -> APP(app(and, x), z)
APP(app(and, app(app(or, y), z)), x) -> APP(app(and, x), y)
APP(app(and, x), app(app(or, y), z)) -> APP(app(and, x), z)
APP(app(and, x), app(app(or, y), z)) -> APP(app(and, x), y)

none


It is proper and hence, it can be A-transformed which results in the DP problem

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)

none


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):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>2
2=1
{1, 2, 3, 4} , {1, 2, 3, 4}
1=1
2>2

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

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


Rules:


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





The original DP problem is in applicative form. Its DPs and usable rules are the following.

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

none


It is proper and hence, it can be A-transformed which results in the DP problem

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

none


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):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1

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