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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Polynomial Ordering


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


Additionally, the following usable rules 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))
app(app(*, app(app(+, y), z)), x) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) -> app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) -> app(app(+, x), app(app(+, y), 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
Polo
           →DP Problem 2
Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

APP(app(*, app(app(*, x), y0)), app(app(+, y''), z'')) -> APP(app(*, x), app(app(+, app(app(*, y0), y'')), app(app(*, y0), z'')))
APP(app(*, app(app(*, x), app(app(+, y''), z'''))), z'') -> APP(app(*, x), app(app(+, app(app(*, z''), y'')), app(app(*, z''), z''')))
APP(app(*, app(app(*, x), app(app(*, x''), y''))), z'') -> APP(app(*, x), app(app(*, x''), app(app(*, y''), z'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 2
Nar
             ...
               →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(*, app(app(*, x), app(app(*, x''), y''))), z'') -> APP(app(*, x), app(app(*, x''), app(app(*, y''), z'')))
APP(app(*, app(app(*, x), y0)), app(app(+, y''), z'')) -> APP(app(*, x), app(app(+, app(app(*, y0), y'')), app(app(*, y0), z'')))
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, y), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(+, app(app(+, x), y)), z) -> APP(app(+, y), z)


Rules:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes