YES

Prover = TRS(tech=PATTERN_RULES, nb_unfoldings=unlimited, max_nb_unfolded_rules=200)

** BEGIN proof argument **
All the DP problems were proved finite.
As all the involved DP processors are sound,
the TRS under analysis terminates.
** END proof argument **

** BEGIN proof description **
## Searching for a generalized rewrite rule
(a rule whose right-hand side contains a variable
that does not occur in the left-hand side)...
No generalized rewrite rule found!

## Applying the DP framework...
## Round 1:
  ## DP problem: 
  Dependency pairs = [mark^#(inf(_0)) -> mark^#(_0), mark^#(take(_0,_1)) -> mark^#(_0), mark^#(take(_0,_1)) -> mark^#(_1), mark^#(length(_0)) -> mark^#(_0)]
  TRS = {a__eq(0,0) -> true, a__eq(s(_0),s(_1)) -> a__eq(_0,_1), a__eq(_0,_1) -> false, a__inf(_0) -> cons(_0,inf(s(_0))), a__take(0,_0) -> nil, a__take(s(_0),cons(_1,_2)) -> cons(_1,take(_0,_2)), a__length(nil) -> 0, a__length(cons(_0,_1)) -> s(length(_1)), mark(eq(_0,_1)) -> a__eq(_0,_1), mark(inf(_0)) -> a__inf(mark(_0)), mark(take(_0,_1)) -> a__take(mark(_0),mark(_1)), mark(length(_0)) -> a__length(mark(_0)), mark(0) -> 0, mark(true) -> true, mark(s(_0)) -> s(_0), mark(false) -> false, mark(cons(_0,_1)) -> cons(_0,_1), mark(nil) -> nil, a__eq(_0,_1) -> eq(_0,_1), a__inf(_0) -> inf(_0), a__take(_0,_1) -> take(_0,_1), a__length(_0) -> length(_0)}
    ## Trying with homeomorphic embeddings... Success!
  This DP problem is finite.
  ## DP problem: 
  Dependency pairs = [a__eq^#(s(_0),s(_1)) -> a__eq^#(_0,_1)]
  TRS = {a__eq(0,0) -> true, a__eq(s(_0),s(_1)) -> a__eq(_0,_1), a__eq(_0,_1) -> false, a__inf(_0) -> cons(_0,inf(s(_0))), a__take(0,_0) -> nil, a__take(s(_0),cons(_1,_2)) -> cons(_1,take(_0,_2)), a__length(nil) -> 0, a__length(cons(_0,_1)) -> s(length(_1)), mark(eq(_0,_1)) -> a__eq(_0,_1), mark(inf(_0)) -> a__inf(mark(_0)), mark(take(_0,_1)) -> a__take(mark(_0),mark(_1)), mark(length(_0)) -> a__length(mark(_0)), mark(0) -> 0, mark(true) -> true, mark(s(_0)) -> s(_0), mark(false) -> false, mark(cons(_0,_1)) -> cons(_0,_1), mark(nil) -> nil, a__eq(_0,_1) -> eq(_0,_1), a__inf(_0) -> inf(_0), a__take(_0,_1) -> take(_0,_1), a__length(_0) -> length(_0)}
    ## Trying with homeomorphic embeddings... Success!
  This DP problem is finite.

** END proof description **

Proof stopped at iteration 0
Number of unfolded rules generated by this proof = 0
Number of unfolded rules generated by all the parallel proofs = 0