Term Rewriting System R:
[x, y, z]
app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, x), app(app(+, y), z)) -> APP(+, app(app(*, x), y))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))


Rule:


app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Strategy:

innermost




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

APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
two new Dependency Pairs are created:

APP(app(*, x''), app(app(+, app(app(+, y''), z'')), z)) -> APP(app(+, app(app(+, app(app(*, x''), y'')), app(app(*, x''), z''))), app(app(*, x''), z))
APP(app(*, x''), app(app(+, y), app(app(+, y''), z''))) -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(*, x''), app(app(+, y), app(app(+, y''), z''))) -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))
APP(app(*, x''), app(app(+, app(app(+, y''), z'')), z)) -> APP(app(+, app(app(+, app(app(*, x''), y'')), app(app(*, x''), z''))), app(app(*, x''), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)


Rule:


app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Strategy:

innermost




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

APP(app(*, x''), app(app(+, app(app(+, y''), z'')), z)) -> APP(app(+, app(app(+, app(app(*, x''), y'')), app(app(*, x''), z''))), app(app(*, x''), z))
three new Dependency Pairs are created:

APP(app(*, x'''), app(app(+, app(app(+, app(app(+, y'), z''')), z'')), z)) -> APP(app(+, app(app(+, app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z'''))), app(app(*, x'''), z''))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, y''), app(app(+, y'), z'''))), z)) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, y''), z'')), app(app(+, y'), z'''))) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z''))), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))

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(app(*, x'''), app(app(+, app(app(+, y''), z'')), app(app(+, y'), z'''))) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z''))), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))
APP(app(*, x'''), app(app(+, app(app(+, y''), app(app(+, y'), z'''))), z)) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, app(app(+, y'), z''')), z'')), z)) -> APP(app(+, app(app(+, app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z'''))), app(app(*, x'''), z''))), app(app(*, x'''), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x''), app(app(+, y), app(app(+, y''), z''))) -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))


Rule:


app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Strategy:

innermost




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

APP(app(*, x''), app(app(+, y), app(app(+, y''), z''))) -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))
three new Dependency Pairs are created:

APP(app(*, x'''), app(app(+, app(app(+, y'''), z')), app(app(+, y''), z''))) -> APP(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, y), app(app(+, app(app(+, y'''), z')), z''))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, y), app(app(+, y''), app(app(+, y'''), z')))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(*, x'''), app(app(+, y), app(app(+, y''), app(app(+, y'''), z')))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))))
APP(app(*, x'''), app(app(+, y), app(app(+, app(app(+, y'''), z')), z''))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, app(app(+, y'''), z')), app(app(+, y''), z''))) -> APP(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, app(app(+, y''), app(app(+, y'), z'''))), z)) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, app(app(+, y'), z''')), z'')), z)) -> APP(app(+, app(app(+, app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z'''))), app(app(*, x'''), z''))), app(app(*, x'''), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x'''), app(app(+, app(app(+, y''), z'')), app(app(+, y'), z'''))) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z''))), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))


Rule:


app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(*, x'''), app(app(+, y), app(app(+, y''), app(app(+, y'''), z')))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))))
APP(app(*, x'''), app(app(+, y), app(app(+, app(app(+, y'''), z')), z''))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, app(app(+, y'''), z')), app(app(+, y''), z''))) -> APP(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, app(app(+, y''), app(app(+, y'), z'''))), z)) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, app(app(+, y'), z''')), z'')), z)) -> APP(app(+, app(app(+, app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z'''))), app(app(*, x'''), z''))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, y''), z'')), app(app(+, y'), z'''))) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z''))), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))


Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*)=  1  
  POL(app(x1, x2))=  x1  
  POL(+)=  0  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)


Rule:


app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(+)=  1  
  POL(APP(x1, x2))=  1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pair:


Rule:


app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes