YES
(VAR X Y L X1 X2)
(RULES 
a__eq(0,0) -> true
a__eq(s(X),s(Y)) -> a__eq(X,Y)
a__eq(X,Y) -> false
a__inf(X) -> cons(X,inf(s(X)))
a__take(0,X) -> nil
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
a__length(nil) -> 0
a__length(cons(X,L)) -> s(length(L))
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
mark(length(X)) -> a__length(mark(X))
mark(0) -> 0
mark(true) -> true
mark(s(X)) -> s(X)
mark(false) -> false
mark(cons(X1,X2)) -> cons(X1,X2)
mark(nil) -> nil
a__eq(X1,X2) -> eq(X1,X2)
a__inf(X) -> inf(X)
a__take(X1,X2) -> take(X1,X2)
a__length(X) -> length(X)
)

Proving termination of rewriting for Ex1_GL02a_GM:

-> Dependency pairs:
nF_a__eq(s(X),s(Y)) -> nF_a__eq(X,Y)
nF_mark(eq(X1,X2)) -> nF_a__eq(X1,X2)
nF_mark(inf(X)) -> nF_a__inf(mark(X))
nF_mark(inf(X)) -> nF_mark(X)
nF_mark(take(X1,X2)) -> nF_a__take(mark(X1),mark(X2))
nF_mark(take(X1,X2)) -> nF_mark(X1)
nF_mark(take(X1,X2)) -> nF_mark(X2)
nF_mark(length(X)) -> nF_a__length(mark(X))
nF_mark(length(X)) -> nF_mark(X)

-> Proof of termination for Ex1_GL02a_GM_1_1:
-> -> Dependency pairs in cycle:
nF_mark(inf(X)) -> nF_mark(X)
nF_mark(length(X)) -> nF_mark(X)
nF_mark(take(X1,X2)) -> nF_mark(X2)
nF_mark(take(X1,X2)) -> nF_mark(X1)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for Ex1_GL02a_GM_1_2:
-> -> Dependency pairs in cycle:
nF_a__eq(s(X),s(Y)) -> nF_a__eq(X,Y)

Termination proved: Cycles verify subterm criterion.

SETTINGS:
Base ordering: Polynomial ordering
Proof mode: SCCs in DG + 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.