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
         ↳Remaining Obligation(s)




The following remains to be proven:
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



Innermost Termination of R could not be shown.
Duration:
0:02 minutes