Term Rewriting System R:
[x, y]
app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pair:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


Rules:


app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


The following usable rules for innermost w.r.t. to the AFS can be oriented:

app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
0 > id

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(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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