NO

Prover = TRS(tech=GUIDED_UNF_TRIPLES, nb_unfoldings=unlimited, unfold_variables=false, max_nb_coefficients=12, max_nb_unfolded_rules=-1, strategy=LEFTMOST_NE)

** BEGIN proof argument **
The following rule was generated while unfolding the analyzed TRS:
[iteration = 5] activate(n__adx(cons(_0,n__zeros))) -> activate(n__adx(cons(0,n__zeros)))
Let l be the left-hand side and r be the right-hand side of this rule.
Let p = epsilon, theta1 = {} and theta2 = {_0->0}.
We have r|p = activate(n__adx(cons(0,n__zeros))) and theta2(theta1(l)) = theta1(r|p).
Hence, the term theta1(l) = activate(n__adx(cons(_0,n__zeros))) loops w.r.t. the analyzed TRS.
** 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 = [incr^#(cons(_0,_1)) -> activate^#(_1), activate^#(n__incr(_0)) -> incr^#(activate(_0)), adx^#(cons(_0,_1)) -> incr^#(cons(_0,n__adx(activate(_1)))), activate^#(n__adx(_0)) -> adx^#(activate(_0)), activate^#(n__incr(_0)) -> activate^#(_0), activate^#(n__adx(_0)) -> activate^#(_0), adx^#(cons(_0,_1)) -> activate^#(_1)]
  TRS = {incr(nil) -> nil, incr(cons(_0,_1)) -> cons(s(_0),n__incr(activate(_1))), adx(nil) -> nil, adx(cons(_0,_1)) -> incr(cons(_0,n__adx(activate(_1)))), nats -> adx(zeros), zeros -> cons(0,n__zeros), head(cons(_0,_1)) -> _0, tail(cons(_0,_1)) -> activate(_1), incr(_0) -> n__incr(_0), adx(_0) -> n__adx(_0), zeros -> n__zeros, activate(n__incr(_0)) -> incr(activate(_0)), activate(n__adx(_0)) -> adx(activate(_0)), activate(n__zeros) -> zeros, activate(_0) -> _0}
    ## Trying with homeomorphic embeddings... Failed!
    ## Trying with polynomial interpretations... Too many coefficients (26)! Aborting!
    ## Trying with lexicographic path orders... Failed!
    ## Trying to prove nontermination by unfolding the dependency pairs with the rules of the TRS
      # max_depth=4, unfold_variables=false:
        # Iteration 0: nontermination not detected, 7 unfolded rules generated.
        # Iteration 1: nontermination not detected, 33 unfolded rules generated.
        # Iteration 2: nontermination not detected, 74 unfolded rules generated.
        # Iteration 3: nontermination not detected, 188 unfolded rules generated.
        # Iteration 4: nontermination not detected, 376 unfolded rules generated.
        # Iteration 5: nontermination detected, 569 unfolded rules generated.
        Here is the successful unfolding. Let IR be the TRS under analysis.
        L0 = activate^#(n__adx(_0)) -> adx^#(activate(_0)) [trans] is in U_IR^0.
        D = adx^#(cons(_0,_1)) -> incr^#(cons(_0,n__adx(activate(_1)))) is a dependency pair of IR.
        We build a composed triple from L0 and D.
        ==> L1 = [activate^#(n__adx(_0)) -> adx^#(activate(_0)), adx^#(cons(_1,_2)) -> incr^#(cons(_1,n__adx(activate(_2))))] [comp] is in U_IR^1.
        Let p1 = [0].
        We unfold the first rule of L1 forwards at position p1
        with the rule activate(_0) -> _0.
        ==> L2 = activate^#(n__adx(cons(_0,_1))) -> incr^#(cons(_0,n__adx(activate(_1)))) [trans] is in U_IR^2.
        D = incr^#(cons(_0,_1)) -> activate^#(_1) is a dependency pair of IR.
        We build a composed triple from L2 and D.
        ==> L3 = [activate^#(n__adx(cons(_0,_1))) -> incr^#(cons(_0,n__adx(activate(_1)))), incr^#(cons(_2,_3)) -> activate^#(_3)] [comp] is in U_IR^3.
        Let p3 = [0, 1, 0].
        We unfold the first rule of L3 forwards at position p3
        with the rule activate(n__zeros) -> zeros.
        ==> L4 = [activate^#(n__adx(cons(_0,n__zeros))) -> incr^#(cons(_0,n__adx(zeros))), incr^#(cons(_1,_2)) -> activate^#(_2)] [comp] is in U_IR^4.
        Let p4 = [0, 1, 0].
        We unfold the first rule of L4 forwards at position p4
        with the rule zeros -> cons(0,n__zeros).
        ==> L5 = activate^#(n__adx(cons(_0,n__zeros))) -> activate^#(n__adx(cons(0,n__zeros))) [trans] is in U_IR^5.
  This DP problem is infinite.

** END proof description **

Proof stopped at iteration 5
Number of unfolded rules generated by this proof = 1247
Number of unfolded rules generated by all the parallel proofs = 8931