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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
F > c
F > s

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering


Dependency Pair:

F(x, c(y)) -> F(y, y)


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(x, c(y)) -> F(y, y)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
c > F

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 4
             ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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