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
             ↳Remaining Obligation(s)




The following remains to be proven:
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



Innermost Termination of R could not be shown.
Duration:
0:00 minutes