Term Rewriting System R:
[y, x]
app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(ack, 0), y) -> APP(succ, y)
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x)), y) -> APP(ack, x)
APP(app(ack, app(succ, x)), y) -> APP(succ, 0)
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, x), app(app(ack, app(succ, x)), y))
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(ack, x)
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, x), app(app(ack, app(succ, x)), y))
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))


Rules:


app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))


Strategy:

innermost




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

APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, x), app(app(ack, app(succ, x)), y))
two new Dependency Pairs are created:

APP(app(ack, app(succ, x'')), app(succ, y'')) -> APP(app(ack, x''), app(app(ack, x''), app(succ, 0)))
APP(app(ack, app(succ, x'')), app(succ, app(succ, y''))) -> APP(app(ack, x''), app(app(ack, x''), app(app(ack, app(succ, x'')), y'')))

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(ack, app(succ, x'')), app(succ, app(succ, y''))) -> APP(app(ack, x''), app(app(ack, x''), app(app(ack, app(succ, x'')), y'')))
APP(app(ack, app(succ, x'')), app(succ, y'')) -> APP(app(ack, x''), app(app(ack, x''), app(succ, 0)))
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)


Rules:


app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))


Strategy:

innermost



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