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

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x), y) -> F(f(x, y), y)
F(s(x), y) -> F(x, y)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering


Dependency Pairs:

F(s(x), y) -> F(x, y)
F(s(x), y) -> F(f(x, y), y)


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(s(x), y) -> F(x, y)
F(s(x), y) -> F(f(x, y), y)


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

f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
s(x1) -> s(x1)
f(x1, x2) -> x1


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


Dependency Pair:


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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