(VAR X XS N)
(STRATEGY CONTEXTSENSITIVE 
  (from 1)
  (cons 1)
  (s 1)
  (head 1)
  (2nd 1)
  (take 1 2)
  (0)
  (nil)
  (sel 1 2)
)
(RULES 
from(X) -> cons(X,from(s(X)))
head(cons(X,XS)) -> X
2nd(cons(X,XS)) -> head(XS)
take(0,XS) -> nil
take(s(N),cons(X,XS)) -> cons(X,take(N,XS))
sel(0,cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,XS)
)

The TRS is orthogonal, thus termination of innermost context-sensitive rewriting is equivalent to termination of context-sensitive rewriting.

Proving termination of context-sensitive rewriting for Ex7_BLR02:

-> Dependency pairs:
nF_2nd(cons(X,XS)) -> nF_head(XS)
nF_2nd(cons(X,XS)) -> XS
nF_sel(s(N),cons(X,XS)) -> nF_sel(N,XS)
nF_sel(s(N),cons(X,XS)) -> XS

-> Proof of termination for Ex7_BLR02_1:
-> -> Dependency pairs in cycle:
nF_sel(s(N),cons(X,XS)) -> nF_sel(N,XS)

Termination proved: Cycles verify subterm criterion.

SETTINGS:
Base ordering: Polynomial ordering
Proof mode: SCCs in CSDG + base ordering
Upper bound for coeffs: 1
Rationals below 1 for all non-replacing args: No
Polynomial interpretation: Linear
Coeffs in polynomials: No rationals
Delta: automatic



Termination was proved succesfully.