Term Rewriting System R:
[X]
f(0) -> cons(0, nf(s(0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
activate(nf(X)) -> f(X)
activate(X) -> X

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(0)) -> F(p(s(0)))
F(s(0)) -> P(s(0))
ACTIVATE(nf(X)) -> F(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pair:

F(s(0)) -> F(p(s(0)))


Rules:


f(0) -> cons(0, nf(s(0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
activate(nf(X)) -> f(X)
activate(X) -> X


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(s(0)) -> F(p(s(0)))


Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

p(s(0)) -> 0


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(s(x1))=  1  
  POL(F(x1))=  x1  
  POL(p(x1))=  0  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


f(0) -> cons(0, nf(s(0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
activate(nf(X)) -> f(X)
activate(X) -> X


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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