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 one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


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




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(not, app(app(or, x), y)) -> APP(app(and, app(not, x)), app(not, y))
six new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

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


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




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(not, app(app(and, x), y)) -> APP(app(or, app(not, x)), app(not, y))
six new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

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




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(and, x), app(app(or, y), z)) -> APP(app(or, app(app(and, x), y)), app(app(and, x), z))
four new Dependency Pairs are created:

APP(app(and, x''), app(app(or, app(app(or, y''), z'')), z)) -> APP(app(or, app(app(or, app(app(and, x''), y'')), app(app(and, x''), z''))), app(app(and, x''), z))
APP(app(and, app(app(or, y''), z'')), app(app(or, y0), z)) -> APP(app(or, app(app(or, app(app(and, y0), y'')), app(app(and, y0), z''))), app(app(and, app(app(or, y''), z'')), z))
APP(app(and, x''), app(app(or, y), app(app(or, y''), z''))) -> APP(app(or, app(app(and, x''), y)), app(app(or, app(app(and, x''), y'')), app(app(and, x''), z'')))
APP(app(and, app(app(or, y''), z''')), app(app(or, y), z'')) -> APP(app(or, app(app(and, app(app(or, y''), z''')), y)), app(app(or, app(app(and, z''), y'')), app(app(and, z''), z''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(and, app(app(or, y''), z''')), app(app(or, y), z'')) -> APP(app(or, app(app(and, app(app(or, y''), z''')), y)), app(app(or, app(app(and, z''), y'')), app(app(and, z''), z''')))
APP(app(and, x''), app(app(or, y), app(app(or, y''), z''))) -> APP(app(or, app(app(and, x''), y)), app(app(or, app(app(and, x''), y'')), app(app(and, x''), z'')))
APP(app(and, app(app(or, y''), z'')), app(app(or, y0), z)) -> APP(app(or, app(app(or, app(app(and, y0), y'')), app(app(and, y0), z''))), app(app(and, app(app(or, y''), z'')), z))
APP(app(and, x''), app(app(or, app(app(or, y''), z'')), z)) -> APP(app(or, app(app(or, app(app(and, x''), y'')), app(app(and, x''), z''))), app(app(and, x''), z))
APP(not, app(app(and, x), app(not, x''))) -> APP(app(or, app(not, x)), x'')
APP(not, app(app(and, app(app(and, x''), y'')), y)) -> APP(app(or, app(app(or, app(not, x'')), app(not, y''))), app(not, y))
APP(not, app(app(or, x), app(app(and, x''), y''))) -> APP(app(and, app(not, x)), app(app(or, app(not, x'')), app(not, y'')))
APP(not, app(app(or, x), app(app(or, x''), y''))) -> APP(app(and, app(not, x)), app(app(and, app(not, x'')), app(not, y'')))
APP(not, app(app(or, x), app(not, x''))) -> APP(app(and, app(not, x)), x'')
APP(not, app(app(or, app(app(and, x''), y'')), y)) -> APP(app(and, app(app(or, app(not, x'')), app(not, y''))), app(not, y))
APP(not, app(app(or, app(app(or, x''), y'')), y)) -> APP(app(and, app(app(and, app(not, x'')), app(not, y''))), app(not, y))
APP(not, app(app(or, app(not, x'')), y)) -> APP(app(and, x''), app(not, y))
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, app(app(or, y), z)), x) -> APP(app(or, app(app(and, x), y)), app(app(and, x), z))
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)
APP(not, app(app(and, app(not, x'')), y)) -> APP(app(or, x''), app(not, y))
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)
APP(not, app(app(and, x), app(app(and, x''), y''))) -> APP(app(or, app(not, x)), app(app(or, app(not, x'')), app(not, 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




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(and, app(app(or, y), z)), x) -> APP(app(or, app(app(and, x), y)), app(app(and, x), z))
four new Dependency Pairs are created:

APP(app(and, app(app(or, app(app(or, y''), z'')), z)), x'') -> APP(app(or, app(app(or, app(app(and, x''), y'')), app(app(and, x''), z''))), app(app(and, x''), z))
APP(app(and, app(app(or, y0), z)), app(app(or, y''), z'')) -> APP(app(or, app(app(or, app(app(and, y0), y'')), app(app(and, y0), z''))), app(app(and, app(app(or, y''), z'')), z))
APP(app(and, app(app(or, y), app(app(or, y''), z''))), x'') -> APP(app(or, app(app(and, x''), y)), app(app(or, app(app(and, x''), y'')), app(app(and, x''), z'')))
APP(app(and, app(app(or, y), z'')), app(app(or, y''), z''')) -> APP(app(or, app(app(and, app(app(or, y''), z''')), y)), app(app(or, app(app(and, z''), y'')), app(app(and, z''), z''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(and, app(app(or, y), z'')), app(app(or, y''), z''')) -> APP(app(or, app(app(and, app(app(or, y''), z''')), y)), app(app(or, app(app(and, z''), y'')), app(app(and, z''), z''')))
APP(app(and, app(app(or, y0), z)), app(app(or, y''), z'')) -> APP(app(or, app(app(or, app(app(and, y0), y'')), app(app(and, y0), z''))), app(app(and, app(app(or, y''), z'')), z))
APP(app(and, x''), app(app(or, y), app(app(or, y''), z''))) -> APP(app(or, app(app(and, x''), y)), app(app(or, app(app(and, x''), y'')), app(app(and, x''), z'')))
APP(app(and, app(app(or, y''), z'')), app(app(or, y0), z)) -> APP(app(or, app(app(or, app(app(and, y0), y'')), app(app(and, y0), z''))), app(app(and, app(app(or, y''), z'')), z))
APP(app(and, x''), app(app(or, app(app(or, y''), z'')), z)) -> APP(app(or, app(app(or, app(app(and, x''), y'')), app(app(and, x''), z''))), app(app(and, x''), z))
APP(not, app(app(and, x), app(app(and, x''), y''))) -> APP(app(or, app(not, x)), app(app(or, app(not, x'')), app(not, y'')))
APP(app(and, app(app(or, y), app(app(or, y''), z''))), x'') -> APP(app(or, app(app(and, x''), y)), app(app(or, app(app(and, x''), y'')), app(app(and, x''), z'')))
APP(not, app(app(and, x), app(not, x''))) -> APP(app(or, app(not, x)), x'')
APP(not, app(app(and, app(app(and, x''), y'')), y)) -> APP(app(or, app(app(or, app(not, x'')), app(not, y''))), app(not, y))
APP(not, app(app(or, x), app(app(and, x''), y''))) -> APP(app(and, app(not, x)), app(app(or, app(not, x'')), app(not, y'')))
APP(not, app(app(or, x), app(app(or, x''), y''))) -> APP(app(and, app(not, x)), app(app(and, app(not, x'')), app(not, y'')))
APP(app(and, app(app(or, app(app(or, y''), z'')), z)), x'') -> APP(app(or, app(app(or, app(app(and, x''), y'')), app(app(and, x''), z''))), app(app(and, x''), z))
APP(not, app(app(or, x), app(not, x''))) -> APP(app(and, app(not, x)), x'')
APP(not, app(app(or, app(app(and, x''), y'')), y)) -> APP(app(and, app(app(or, app(not, x'')), app(not, y''))), app(not, y))
APP(not, app(app(or, app(app(or, x''), y'')), y)) -> APP(app(and, app(app(and, app(not, x'')), app(not, y''))), app(not, y))
APP(not, app(app(or, app(not, x'')), y)) -> APP(app(and, x''), app(not, y))
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)
APP(not, app(app(and, app(not, x'')), y)) -> APP(app(or, x''), app(not, y))
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)
APP(app(and, app(app(or, y''), z''')), app(app(or, y), z'')) -> APP(app(or, app(app(and, app(app(or, y''), z''')), y)), app(app(or, app(app(and, z''), y'')), app(app(and, z''), z''')))


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



Innermost Termination of R could not be shown.
Duration:
0:38 minutes