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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(:, app(app(:, x), z))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(:, x)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, app(app(:, x), y)), z)), u)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(:, app(app(:, app(app(:, x), y)), z))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), y)), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(:, app(app(:, x), y))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), y)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), y)), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, app(app(:, x), y)), z)), u)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))


Rule:


app(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> app(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))


Strategy:

innermost



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