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
         ↳Argument Filtering and Ordering


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




The following dependency pairs can be strictly oriented:

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))


The following usable rules for innermost can be oriented:

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))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{app, 0, APP} > succ

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)
app(x1, x2) -> app(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


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




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes