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))

Innermost 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 three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules


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))


Strategy:

innermost




As we are in the innermost case, we can delete all 5 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 4
             ↳A-Transformation
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules


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)


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 4
             ↳ATrans
             ...
               →DP Problem 5
                 ↳Size-Change Principle
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules


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)


Rule:

none


Strategy:

innermost




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
         ↳UsableRules
       →DP Problem 2
         ↳Usable Rules (Innermost)
       →DP Problem 3
         ↳UsableRules


Dependency Pairs:

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))


Strategy:

innermost




As we are in the innermost case, we can delete all 5 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 6
             ↳A-Transformation
       →DP Problem 3
         ↳UsableRules


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 6
             ↳ATrans
             ...
               →DP Problem 7
                 ↳Size-Change Principle
       →DP Problem 3
         ↳UsableRules


Dependency Pairs:

NOT(or(x, y)) -> NOT(y)
NOT(or(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)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {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:
or(x1, x2) -> or(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳Usable Rules (Innermost)


Dependency Pairs:

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


Strategy:

innermost




As we are in the innermost case, we can delete all 5 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
           →DP Problem 8
             ↳A-Transformation


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
           →DP Problem 8
             ↳ATrans
             ...
               →DP Problem 9
                 ↳Size-Change Principle


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




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

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

We obtain no new DP problems.

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