YES

Problem:
 active(nats()) -> mark(adx(zeros()))
 active(zeros()) -> mark(cons(0(),zeros()))
 active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
 active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
 active(hd(cons(X,Y))) -> mark(X)
 active(tl(cons(X,Y))) -> mark(Y)
 active(adx(X)) -> adx(active(X))
 active(incr(X)) -> incr(active(X))
 active(hd(X)) -> hd(active(X))
 active(tl(X)) -> tl(active(X))
 adx(mark(X)) -> mark(adx(X))
 incr(mark(X)) -> mark(incr(X))
 hd(mark(X)) -> mark(hd(X))
 tl(mark(X)) -> mark(tl(X))
 proper(nats()) -> ok(nats())
 proper(adx(X)) -> adx(proper(X))
 proper(zeros()) -> ok(zeros())
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 proper(incr(X)) -> incr(proper(X))
 proper(s(X)) -> s(proper(X))
 proper(hd(X)) -> hd(proper(X))
 proper(tl(X)) -> tl(proper(X))
 adx(ok(X)) -> ok(adx(X))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 incr(ok(X)) -> ok(incr(X))
 s(ok(X)) -> ok(s(X))
 hd(ok(X)) -> ok(hd(X))
 tl(ok(X)) -> ok(tl(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [top](x0) = x0 + 4,
   
   [ok](x0) = 4x0,
   
   [proper](x0) = 4x0,
   
   [tl](x0) = x0,
   
   [hd](x0) = x0,
   
   [s](x0) = x0,
   
   [incr](x0) = x0,
   
   [cons](x0, x1) = 4x0 + 4x1,
   
   [0] = 0,
   
   [mark](x0) = 4x0,
   
   [adx](x0) = x0,
   
   [zeros] = 0,
   
   [active](x0) = 4x0,
   
   [nats] = 2
  orientation:
   active(nats()) = 8 >= 0 = mark(adx(zeros()))
   
   active(zeros()) = 0 >= 0 = mark(cons(0(),zeros()))
   
   active(incr(cons(X,Y))) = 16X + 16Y >= 16X + 16Y = mark(cons(s(X),incr(Y)))
   
   active(adx(cons(X,Y))) = 16X + 16Y >= 16X + 16Y = mark(incr(cons(X,adx(Y))))
   
   active(hd(cons(X,Y))) = 16X + 16Y >= 4X = mark(X)
   
   active(tl(cons(X,Y))) = 16X + 16Y >= 4Y = mark(Y)
   
   active(adx(X)) = 4X >= 4X = adx(active(X))
   
   active(incr(X)) = 4X >= 4X = incr(active(X))
   
   active(hd(X)) = 4X >= 4X = hd(active(X))
   
   active(tl(X)) = 4X >= 4X = tl(active(X))
   
   adx(mark(X)) = 4X >= 4X = mark(adx(X))
   
   incr(mark(X)) = 4X >= 4X = mark(incr(X))
   
   hd(mark(X)) = 4X >= 4X = mark(hd(X))
   
   tl(mark(X)) = 4X >= 4X = mark(tl(X))
   
   proper(nats()) = 8 >= 8 = ok(nats())
   
   proper(adx(X)) = 4X >= 4X = adx(proper(X))
   
   proper(zeros()) = 0 >= 0 = ok(zeros())
   
   proper(cons(X1,X2)) = 16X1 + 16X2 >= 16X1 + 16X2 = cons(proper(X1),proper(X2))
   
   proper(0()) = 0 >= 0 = ok(0())
   
   proper(incr(X)) = 4X >= 4X = incr(proper(X))
   
   proper(s(X)) = 4X >= 4X = s(proper(X))
   
   proper(hd(X)) = 4X >= 4X = hd(proper(X))
   
   proper(tl(X)) = 4X >= 4X = tl(proper(X))
   
   adx(ok(X)) = 4X >= 4X = ok(adx(X))
   
   cons(ok(X1),ok(X2)) = 16X1 + 16X2 >= 16X1 + 16X2 = ok(cons(X1,X2))
   
   incr(ok(X)) = 4X >= 4X = ok(incr(X))
   
   s(ok(X)) = 4X >= 4X = ok(s(X))
   
   hd(ok(X)) = 4X >= 4X = ok(hd(X))
   
   tl(ok(X)) = 4X >= 4X = ok(tl(X))
   
   top(mark(X)) = 4X + 4 >= 4X + 4 = top(proper(X))
   
   top(ok(X)) = 4X + 4 >= 4X + 4 = top(active(X))
  problem:
   active(zeros()) -> mark(cons(0(),zeros()))
   active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
   active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
   active(hd(cons(X,Y))) -> mark(X)
   active(tl(cons(X,Y))) -> mark(Y)
   active(adx(X)) -> adx(active(X))
   active(incr(X)) -> incr(active(X))
   active(hd(X)) -> hd(active(X))
   active(tl(X)) -> tl(active(X))
   adx(mark(X)) -> mark(adx(X))
   incr(mark(X)) -> mark(incr(X))
   hd(mark(X)) -> mark(hd(X))
   tl(mark(X)) -> mark(tl(X))
   proper(nats()) -> ok(nats())
   proper(adx(X)) -> adx(proper(X))
   proper(zeros()) -> ok(zeros())
   proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   proper(incr(X)) -> incr(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(hd(X)) -> hd(proper(X))
   proper(tl(X)) -> tl(proper(X))
   adx(ok(X)) -> ok(adx(X))
   cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
   incr(ok(X)) -> ok(incr(X))
   s(ok(X)) -> ok(s(X))
   hd(ok(X)) -> ok(hd(X))
   tl(ok(X)) -> ok(tl(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  DP Processor:
   DPs:
    active#(zeros()) -> cons#(0(),zeros())
    active#(incr(cons(X,Y))) -> incr#(Y)
    active#(incr(cons(X,Y))) -> s#(X)
    active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y))
    active#(adx(cons(X,Y))) -> adx#(Y)
    active#(adx(cons(X,Y))) -> cons#(X,adx(Y))
    active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y)))
    active#(adx(X)) -> active#(X)
    active#(adx(X)) -> adx#(active(X))
    active#(incr(X)) -> active#(X)
    active#(incr(X)) -> incr#(active(X))
    active#(hd(X)) -> active#(X)
    active#(hd(X)) -> hd#(active(X))
    active#(tl(X)) -> active#(X)
    active#(tl(X)) -> tl#(active(X))
    adx#(mark(X)) -> adx#(X)
    incr#(mark(X)) -> incr#(X)
    hd#(mark(X)) -> hd#(X)
    tl#(mark(X)) -> tl#(X)
    proper#(adx(X)) -> proper#(X)
    proper#(adx(X)) -> adx#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(incr(X)) -> proper#(X)
    proper#(incr(X)) -> incr#(proper(X))
    proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> s#(proper(X))
    proper#(hd(X)) -> proper#(X)
    proper#(hd(X)) -> hd#(proper(X))
    proper#(tl(X)) -> proper#(X)
    proper#(tl(X)) -> tl#(proper(X))
    adx#(ok(X)) -> adx#(X)
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    incr#(ok(X)) -> incr#(X)
    s#(ok(X)) -> s#(X)
    hd#(ok(X)) -> hd#(X)
    tl#(ok(X)) -> tl#(X)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    active(zeros()) -> mark(cons(0(),zeros()))
    active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
    active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
    active(hd(cons(X,Y))) -> mark(X)
    active(tl(cons(X,Y))) -> mark(Y)
    active(adx(X)) -> adx(active(X))
    active(incr(X)) -> incr(active(X))
    active(hd(X)) -> hd(active(X))
    active(tl(X)) -> tl(active(X))
    adx(mark(X)) -> mark(adx(X))
    incr(mark(X)) -> mark(incr(X))
    hd(mark(X)) -> mark(hd(X))
    tl(mark(X)) -> mark(tl(X))
    proper(nats()) -> ok(nats())
    proper(adx(X)) -> adx(proper(X))
    proper(zeros()) -> ok(zeros())
    proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    proper(incr(X)) -> incr(proper(X))
    proper(s(X)) -> s(proper(X))
    proper(hd(X)) -> hd(proper(X))
    proper(tl(X)) -> tl(proper(X))
    adx(ok(X)) -> ok(adx(X))
    cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    incr(ok(X)) -> ok(incr(X))
    s(ok(X)) -> ok(s(X))
    hd(ok(X)) -> ok(hd(X))
    tl(ok(X)) -> ok(tl(X))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   TDG Processor:
    DPs:
     active#(zeros()) -> cons#(0(),zeros())
     active#(incr(cons(X,Y))) -> incr#(Y)
     active#(incr(cons(X,Y))) -> s#(X)
     active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y))
     active#(adx(cons(X,Y))) -> adx#(Y)
     active#(adx(cons(X,Y))) -> cons#(X,adx(Y))
     active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y)))
     active#(adx(X)) -> active#(X)
     active#(adx(X)) -> adx#(active(X))
     active#(incr(X)) -> active#(X)
     active#(incr(X)) -> incr#(active(X))
     active#(hd(X)) -> active#(X)
     active#(hd(X)) -> hd#(active(X))
     active#(tl(X)) -> active#(X)
     active#(tl(X)) -> tl#(active(X))
     adx#(mark(X)) -> adx#(X)
     incr#(mark(X)) -> incr#(X)
     hd#(mark(X)) -> hd#(X)
     tl#(mark(X)) -> tl#(X)
     proper#(adx(X)) -> proper#(X)
     proper#(adx(X)) -> adx#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(incr(X)) -> proper#(X)
     proper#(incr(X)) -> incr#(proper(X))
     proper#(s(X)) -> proper#(X)
     proper#(s(X)) -> s#(proper(X))
     proper#(hd(X)) -> proper#(X)
     proper#(hd(X)) -> hd#(proper(X))
     proper#(tl(X)) -> proper#(X)
     proper#(tl(X)) -> tl#(proper(X))
     adx#(ok(X)) -> adx#(X)
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     incr#(ok(X)) -> incr#(X)
     s#(ok(X)) -> s#(X)
     hd#(ok(X)) -> hd#(X)
     tl#(ok(X)) -> tl#(X)
     top#(mark(X)) -> proper#(X)
     top#(mark(X)) -> top#(proper(X))
     top#(ok(X)) -> active#(X)
     top#(ok(X)) -> top#(active(X))
    TRS:
     active(zeros()) -> mark(cons(0(),zeros()))
     active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
     active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
     active(hd(cons(X,Y))) -> mark(X)
     active(tl(cons(X,Y))) -> mark(Y)
     active(adx(X)) -> adx(active(X))
     active(incr(X)) -> incr(active(X))
     active(hd(X)) -> hd(active(X))
     active(tl(X)) -> tl(active(X))
     adx(mark(X)) -> mark(adx(X))
     incr(mark(X)) -> mark(incr(X))
     hd(mark(X)) -> mark(hd(X))
     tl(mark(X)) -> mark(tl(X))
     proper(nats()) -> ok(nats())
     proper(adx(X)) -> adx(proper(X))
     proper(zeros()) -> ok(zeros())
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(incr(X)) -> incr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(hd(X)) -> hd(proper(X))
     proper(tl(X)) -> tl(proper(X))
     adx(ok(X)) -> ok(adx(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     incr(ok(X)) -> ok(incr(X))
     s(ok(X)) -> ok(s(X))
     hd(ok(X)) -> ok(hd(X))
     tl(ok(X)) -> ok(tl(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    graph:
     top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X))
     top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
     top#(ok(X)) -> top#(active(X)) ->
     top#(mark(X)) -> top#(proper(X))
     top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X)
     top#(ok(X)) -> active#(X) -> active#(tl(X)) -> tl#(active(X))
     top#(ok(X)) -> active#(X) -> active#(tl(X)) -> active#(X)
     top#(ok(X)) -> active#(X) -> active#(hd(X)) -> hd#(active(X))
     top#(ok(X)) -> active#(X) -> active#(hd(X)) -> active#(X)
     top#(ok(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X))
     top#(ok(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     top#(ok(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X))
     top#(ok(X)) -> active#(X) -> active#(adx(X)) -> active#(X)
     top#(ok(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y)))
     top#(ok(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> cons#(X,adx(Y))
     top#(ok(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> adx#(Y)
     top#(ok(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y))
     top#(ok(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> s#(X)
     top#(ok(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> incr#(Y)
     top#(ok(X)) -> active#(X) ->
     active#(zeros()) -> cons#(0(),zeros())
     top#(mark(X)) -> top#(proper(X)) ->
     top#(ok(X)) -> top#(active(X))
     top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X)
     top#(mark(X)) -> top#(proper(X)) ->
     top#(mark(X)) -> top#(proper(X))
     top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X)
     top#(mark(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X))
     top#(mark(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X)
     top#(mark(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X))
     top#(mark(X)) -> proper#(X) -> proper#(hd(X)) -> proper#(X)
     top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
     top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
     top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X))
     top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
     top#(mark(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1)
     top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2)
     top#(mark(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X))
     top#(mark(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X)
     proper#(tl(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X))
     proper#(tl(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X)
     proper#(tl(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X))
     proper#(tl(X)) -> proper#(X) -> proper#(hd(X)) -> proper#(X)
     proper#(tl(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
     proper#(tl(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
     proper#(tl(X)) -> proper#(X) ->
     proper#(incr(X)) -> incr#(proper(X))
     proper#(tl(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
     proper#(tl(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(tl(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(tl(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(tl(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X))
     proper#(tl(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X)
     proper#(tl(X)) -> tl#(proper(X)) -> tl#(ok(X)) -> tl#(X)
     proper#(tl(X)) -> tl#(proper(X)) -> tl#(mark(X)) -> tl#(X)
     proper#(hd(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X))
     proper#(hd(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X)
     proper#(hd(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X))
     proper#(hd(X)) -> proper#(X) -> proper#(hd(X)) -> proper#(X)
     proper#(hd(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
     proper#(hd(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
     proper#(hd(X)) -> proper#(X) ->
     proper#(incr(X)) -> incr#(proper(X))
     proper#(hd(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
     proper#(hd(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(hd(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(hd(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(hd(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X))
     proper#(hd(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X)
     proper#(hd(X)) -> hd#(proper(X)) -> hd#(ok(X)) -> hd#(X)
     proper#(hd(X)) -> hd#(proper(X)) -> hd#(mark(X)) -> hd#(X)
     proper#(s(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X))
     proper#(s(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X)
     proper#(s(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X))
     proper#(s(X)) -> proper#(X) -> proper#(hd(X)) -> proper#(X)
     proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
     proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
     proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X))
     proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
     proper#(s(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1)
     proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2)
     proper#(s(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X))
     proper#(s(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X)
     proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X)
     proper#(incr(X)) -> proper#(X) ->
     proper#(tl(X)) -> tl#(proper(X))
     proper#(incr(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X)
     proper#(incr(X)) -> proper#(X) ->
     proper#(hd(X)) -> hd#(proper(X))
     proper#(incr(X)) -> proper#(X) -> proper#(hd(X)) -> proper#(X)
     proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
     proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
     proper#(incr(X)) -> proper#(X) ->
     proper#(incr(X)) -> incr#(proper(X))
     proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
     proper#(incr(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(incr(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(incr(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(incr(X)) -> proper#(X) ->
     proper#(adx(X)) -> adx#(proper(X))
     proper#(incr(X)) -> proper#(X) ->
     proper#(adx(X)) -> proper#(X)
     proper#(incr(X)) -> incr#(proper(X)) ->
     incr#(ok(X)) -> incr#(X)
     proper#(incr(X)) -> incr#(proper(X)) ->
     incr#(mark(X)) -> incr#(X)
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(tl(X)) -> tl#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(tl(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(hd(X)) -> hd#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(hd(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(s(X)) -> s#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(s(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(incr(X)) -> incr#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(incr(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(adx(X)) -> adx#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X2) ->
     proper#(adx(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(tl(X)) -> tl#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(tl(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(hd(X)) -> hd#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(hd(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(s(X)) -> s#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(s(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(incr(X)) -> incr#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(incr(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(adx(X)) -> adx#(proper(X))
     proper#(cons(X1,X2)) -> proper#(X1) ->
     proper#(adx(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) ->
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     proper#(adx(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X))
     proper#(adx(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X)
     proper#(adx(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X))
     proper#(adx(X)) -> proper#(X) -> proper#(hd(X)) -> proper#(X)
     proper#(adx(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
     proper#(adx(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
     proper#(adx(X)) -> proper#(X) ->
     proper#(incr(X)) -> incr#(proper(X))
     proper#(adx(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
     proper#(adx(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(adx(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(adx(X)) -> proper#(X) ->
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(adx(X)) -> proper#(X) ->
     proper#(adx(X)) -> adx#(proper(X))
     proper#(adx(X)) -> proper#(X) ->
     proper#(adx(X)) -> proper#(X)
     proper#(adx(X)) -> adx#(proper(X)) -> adx#(ok(X)) -> adx#(X)
     proper#(adx(X)) -> adx#(proper(X)) -> adx#(mark(X)) -> adx#(X)
     tl#(ok(X)) -> tl#(X) -> tl#(ok(X)) -> tl#(X)
     tl#(ok(X)) -> tl#(X) -> tl#(mark(X)) -> tl#(X)
     tl#(mark(X)) -> tl#(X) -> tl#(ok(X)) -> tl#(X)
     tl#(mark(X)) -> tl#(X) -> tl#(mark(X)) -> tl#(X)
     hd#(ok(X)) -> hd#(X) -> hd#(ok(X)) -> hd#(X)
     hd#(ok(X)) -> hd#(X) -> hd#(mark(X)) -> hd#(X)
     hd#(mark(X)) -> hd#(X) -> hd#(ok(X)) -> hd#(X)
     hd#(mark(X)) -> hd#(X) -> hd#(mark(X)) -> hd#(X)
     adx#(ok(X)) -> adx#(X) -> adx#(ok(X)) -> adx#(X)
     adx#(ok(X)) -> adx#(X) -> adx#(mark(X)) -> adx#(X)
     adx#(mark(X)) -> adx#(X) -> adx#(ok(X)) -> adx#(X)
     adx#(mark(X)) -> adx#(X) -> adx#(mark(X)) -> adx#(X)
     s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X)
     incr#(ok(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X)
     incr#(ok(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X)
     incr#(mark(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X)
     incr#(mark(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X)
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2) ->
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     active#(tl(X)) -> tl#(active(X)) -> tl#(ok(X)) -> tl#(X)
     active#(tl(X)) -> tl#(active(X)) -> tl#(mark(X)) -> tl#(X)
     active#(tl(X)) -> active#(X) -> active#(tl(X)) -> tl#(active(X))
     active#(tl(X)) -> active#(X) -> active#(tl(X)) -> active#(X)
     active#(tl(X)) -> active#(X) -> active#(hd(X)) -> hd#(active(X))
     active#(tl(X)) -> active#(X) -> active#(hd(X)) -> active#(X)
     active#(tl(X)) -> active#(X) ->
     active#(incr(X)) -> incr#(active(X))
     active#(tl(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     active#(tl(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X))
     active#(tl(X)) -> active#(X) -> active#(adx(X)) -> active#(X)
     active#(tl(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y)))
     active#(tl(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> cons#(X,adx(Y))
     active#(tl(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> adx#(Y)
     active#(tl(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y))
     active#(tl(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> s#(X)
     active#(tl(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> incr#(Y)
     active#(tl(X)) -> active#(X) ->
     active#(zeros()) -> cons#(0(),zeros())
     active#(hd(X)) -> hd#(active(X)) -> hd#(ok(X)) -> hd#(X)
     active#(hd(X)) -> hd#(active(X)) -> hd#(mark(X)) -> hd#(X)
     active#(hd(X)) -> active#(X) -> active#(tl(X)) -> tl#(active(X))
     active#(hd(X)) -> active#(X) -> active#(tl(X)) -> active#(X)
     active#(hd(X)) -> active#(X) -> active#(hd(X)) -> hd#(active(X))
     active#(hd(X)) -> active#(X) -> active#(hd(X)) -> active#(X)
     active#(hd(X)) -> active#(X) ->
     active#(incr(X)) -> incr#(active(X))
     active#(hd(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     active#(hd(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X))
     active#(hd(X)) -> active#(X) -> active#(adx(X)) -> active#(X)
     active#(hd(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y)))
     active#(hd(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> cons#(X,adx(Y))
     active#(hd(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> adx#(Y)
     active#(hd(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y))
     active#(hd(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> s#(X)
     active#(hd(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> incr#(Y)
     active#(hd(X)) -> active#(X) ->
     active#(zeros()) -> cons#(0(),zeros())
     active#(incr(cons(X,Y))) -> s#(X) -> s#(ok(X)) -> s#(X)
     active#(incr(cons(X,Y))) -> incr#(Y) ->
     incr#(ok(X)) -> incr#(X)
     active#(incr(cons(X,Y))) -> incr#(Y) ->
     incr#(mark(X)) -> incr#(X)
     active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y)) ->
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     active#(incr(X)) -> incr#(active(X)) ->
     incr#(ok(X)) -> incr#(X)
     active#(incr(X)) -> incr#(active(X)) ->
     incr#(mark(X)) -> incr#(X)
     active#(incr(X)) -> active#(X) ->
     active#(tl(X)) -> tl#(active(X))
     active#(incr(X)) -> active#(X) -> active#(tl(X)) -> active#(X)
     active#(incr(X)) -> active#(X) ->
     active#(hd(X)) -> hd#(active(X))
     active#(incr(X)) -> active#(X) -> active#(hd(X)) -> active#(X)
     active#(incr(X)) -> active#(X) ->
     active#(incr(X)) -> incr#(active(X))
     active#(incr(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     active#(incr(X)) -> active#(X) ->
     active#(adx(X)) -> adx#(active(X))
     active#(incr(X)) -> active#(X) -> active#(adx(X)) -> active#(X)
     active#(incr(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y)))
     active#(incr(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> cons#(X,adx(Y))
     active#(incr(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> adx#(Y)
     active#(incr(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y))
     active#(incr(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> s#(X)
     active#(incr(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> incr#(Y)
     active#(incr(X)) -> active#(X) ->
     active#(zeros()) -> cons#(0(),zeros())
     active#(adx(cons(X,Y))) -> adx#(Y) -> adx#(ok(X)) -> adx#(X)
     active#(adx(cons(X,Y))) -> adx#(Y) ->
     adx#(mark(X)) -> adx#(X)
     active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) ->
     incr#(ok(X)) -> incr#(X)
     active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) ->
     incr#(mark(X)) -> incr#(X)
     active#(adx(cons(X,Y))) -> cons#(X,adx(Y)) ->
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     active#(adx(X)) -> adx#(active(X)) -> adx#(ok(X)) -> adx#(X)
     active#(adx(X)) -> adx#(active(X)) -> adx#(mark(X)) -> adx#(X)
     active#(adx(X)) -> active#(X) -> active#(tl(X)) -> tl#(active(X))
     active#(adx(X)) -> active#(X) -> active#(tl(X)) -> active#(X)
     active#(adx(X)) -> active#(X) -> active#(hd(X)) -> hd#(active(X))
     active#(adx(X)) -> active#(X) -> active#(hd(X)) -> active#(X)
     active#(adx(X)) -> active#(X) ->
     active#(incr(X)) -> incr#(active(X))
     active#(adx(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     active#(adx(X)) -> active#(X) ->
     active#(adx(X)) -> adx#(active(X))
     active#(adx(X)) -> active#(X) -> active#(adx(X)) -> active#(X)
     active#(adx(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y)))
     active#(adx(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> cons#(X,adx(Y))
     active#(adx(X)) -> active#(X) ->
     active#(adx(cons(X,Y))) -> adx#(Y)
     active#(adx(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y))
     active#(adx(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> s#(X)
     active#(adx(X)) -> active#(X) ->
     active#(incr(cons(X,Y))) -> incr#(Y)
     active#(adx(X)) -> active#(X) ->
     active#(zeros()) -> cons#(0(),zeros())
     active#(zeros()) -> cons#(0(),zeros()) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    SCC Processor:
     #sccs: 9
     #rules: 23
     #arcs: 233/1764
     DPs:
      top#(ok(X)) -> top#(active(X))
      top#(mark(X)) -> top#(proper(X))
     TRS:
      active(zeros()) -> mark(cons(0(),zeros()))
      active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
      active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
      active(hd(cons(X,Y))) -> mark(X)
      active(tl(cons(X,Y))) -> mark(Y)
      active(adx(X)) -> adx(active(X))
      active(incr(X)) -> incr(active(X))
      active(hd(X)) -> hd(active(X))
      active(tl(X)) -> tl(active(X))
      adx(mark(X)) -> mark(adx(X))
      incr(mark(X)) -> mark(incr(X))
      hd(mark(X)) -> mark(hd(X))
      tl(mark(X)) -> mark(tl(X))
      proper(nats()) -> ok(nats())
      proper(adx(X)) -> adx(proper(X))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(hd(X)) -> hd(proper(X))
      proper(tl(X)) -> tl(proper(X))
      adx(ok(X)) -> ok(adx(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      hd(ok(X)) -> ok(hd(X))
      tl(ok(X)) -> ok(tl(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Usable Rule Processor:
      DPs:
       top#(ok(X)) -> top#(active(X))
       top#(mark(X)) -> top#(proper(X))
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
       active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
       active(hd(cons(X,Y))) -> mark(X)
       active(tl(cons(X,Y))) -> mark(Y)
       active(adx(X)) -> adx(active(X))
       active(incr(X)) -> incr(active(X))
       active(hd(X)) -> hd(active(X))
       active(tl(X)) -> tl(active(X))
       incr(mark(X)) -> mark(incr(X))
       incr(ok(X)) -> ok(incr(X))
       s(ok(X)) -> ok(s(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       adx(mark(X)) -> mark(adx(X))
       adx(ok(X)) -> ok(adx(X))
       hd(mark(X)) -> mark(hd(X))
       hd(ok(X)) -> ok(hd(X))
       tl(mark(X)) -> mark(tl(X))
       tl(ok(X)) -> ok(tl(X))
       proper(nats()) -> ok(nats())
       proper(adx(X)) -> adx(proper(X))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(incr(X)) -> incr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(hd(X)) -> hd(proper(X))
       proper(tl(X)) -> tl(proper(X))
      Matrix Interpretation Processor: dim=1
       
       usable rules:
        active(zeros()) -> mark(cons(0(),zeros()))
        active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
        active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
        active(hd(cons(X,Y))) -> mark(X)
        active(tl(cons(X,Y))) -> mark(Y)
        active(adx(X)) -> adx(active(X))
        active(incr(X)) -> incr(active(X))
        active(hd(X)) -> hd(active(X))
        active(tl(X)) -> tl(active(X))
        incr(mark(X)) -> mark(incr(X))
        incr(ok(X)) -> ok(incr(X))
        s(ok(X)) -> ok(s(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        adx(mark(X)) -> mark(adx(X))
        adx(ok(X)) -> ok(adx(X))
        hd(mark(X)) -> mark(hd(X))
        hd(ok(X)) -> ok(hd(X))
        tl(mark(X)) -> mark(tl(X))
        tl(ok(X)) -> ok(tl(X))
        proper(nats()) -> ok(nats())
        proper(adx(X)) -> adx(proper(X))
        proper(zeros()) -> ok(zeros())
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        proper(incr(X)) -> incr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(hd(X)) -> hd(proper(X))
        proper(tl(X)) -> tl(proper(X))
       interpretation:
        [top#](x0) = 2x0,
        
        [ok](x0) = x0,
        
        [proper](x0) = x0,
        
        [tl](x0) = 2x0 + 2,
        
        [hd](x0) = 3x0 + 3/2,
        
        [s](x0) = 0,
        
        [incr](x0) = x0 + 1,
        
        [cons](x0, x1) = 1/2x0 + 1/2x1,
        
        [0] = 0,
        
        [mark](x0) = x0 + 1/2,
        
        [adx](x0) = x0 + 3,
        
        [zeros] = 1,
        
        [active](x0) = x0,
        
        [nats] = 0
       orientation:
        top#(ok(X)) = 2X >= 2X = top#(active(X))
        
        top#(mark(X)) = 2X + 1 >= 2X = top#(proper(X))
        
        active(zeros()) = 1 >= 1 = mark(cons(0(),zeros()))
        
        active(incr(cons(X,Y))) = 1/2X + 1/2Y + 1 >= 1/2Y + 1 = mark(cons(s(X),incr(Y)))
        
        active(adx(cons(X,Y))) = 1/2X + 1/2Y + 3 >= 1/2X + 1/2Y + 3 = mark(incr(cons(X,adx(Y))))
        
        active(hd(cons(X,Y))) = 3/2X + 3/2Y + 3/2 >= X + 1/2 = mark(X)
        
        active(tl(cons(X,Y))) = X + Y + 2 >= Y + 1/2 = mark(Y)
        
        active(adx(X)) = X + 3 >= X + 3 = adx(active(X))
        
        active(incr(X)) = X + 1 >= X + 1 = incr(active(X))
        
        active(hd(X)) = 3X + 3/2 >= 3X + 3/2 = hd(active(X))
        
        active(tl(X)) = 2X + 2 >= 2X + 2 = tl(active(X))
        
        incr(mark(X)) = X + 3/2 >= X + 3/2 = mark(incr(X))
        
        incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X))
        
        s(ok(X)) = 0 >= 0 = ok(s(X))
        
        cons(ok(X1),ok(X2)) = 1/2X1 + 1/2X2 >= 1/2X1 + 1/2X2 = ok(cons(X1,X2))
        
        adx(mark(X)) = X + 7/2 >= X + 7/2 = mark(adx(X))
        
        adx(ok(X)) = X + 3 >= X + 3 = ok(adx(X))
        
        hd(mark(X)) = 3X + 3 >= 3X + 2 = mark(hd(X))
        
        hd(ok(X)) = 3X + 3/2 >= 3X + 3/2 = ok(hd(X))
        
        tl(mark(X)) = 2X + 3 >= 2X + 5/2 = mark(tl(X))
        
        tl(ok(X)) = 2X + 2 >= 2X + 2 = ok(tl(X))
        
        proper(nats()) = 0 >= 0 = ok(nats())
        
        proper(adx(X)) = X + 3 >= X + 3 = adx(proper(X))
        
        proper(zeros()) = 1 >= 1 = ok(zeros())
        
        proper(cons(X1,X2)) = 1/2X1 + 1/2X2 >= 1/2X1 + 1/2X2 = cons(proper(X1),proper(X2))
        
        proper(0()) = 0 >= 0 = ok(0())
        
        proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X))
        
        proper(s(X)) = 0 >= 0 = s(proper(X))
        
        proper(hd(X)) = 3X + 3/2 >= 3X + 3/2 = hd(proper(X))
        
        proper(tl(X)) = 2X + 2 >= 2X + 2 = tl(proper(X))
       problem:
        DPs:
         top#(ok(X)) -> top#(active(X))
        TRS:
         active(zeros()) -> mark(cons(0(),zeros()))
         active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
         active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
         active(hd(cons(X,Y))) -> mark(X)
         active(tl(cons(X,Y))) -> mark(Y)
         active(adx(X)) -> adx(active(X))
         active(incr(X)) -> incr(active(X))
         active(hd(X)) -> hd(active(X))
         active(tl(X)) -> tl(active(X))
         incr(mark(X)) -> mark(incr(X))
         incr(ok(X)) -> ok(incr(X))
         s(ok(X)) -> ok(s(X))
         cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
         adx(mark(X)) -> mark(adx(X))
         adx(ok(X)) -> ok(adx(X))
         hd(mark(X)) -> mark(hd(X))
         hd(ok(X)) -> ok(hd(X))
         tl(mark(X)) -> mark(tl(X))
         tl(ok(X)) -> ok(tl(X))
         proper(nats()) -> ok(nats())
         proper(adx(X)) -> adx(proper(X))
         proper(zeros()) -> ok(zeros())
         proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
         proper(0()) -> ok(0())
         proper(incr(X)) -> incr(proper(X))
         proper(s(X)) -> s(proper(X))
         proper(hd(X)) -> hd(proper(X))
         proper(tl(X)) -> tl(proper(X))
       Restore Modifier:
        DPs:
         top#(ok(X)) -> top#(active(X))
        TRS:
         active(zeros()) -> mark(cons(0(),zeros()))
         active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
         active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
         active(hd(cons(X,Y))) -> mark(X)
         active(tl(cons(X,Y))) -> mark(Y)
         active(adx(X)) -> adx(active(X))
         active(incr(X)) -> incr(active(X))
         active(hd(X)) -> hd(active(X))
         active(tl(X)) -> tl(active(X))
         adx(mark(X)) -> mark(adx(X))
         incr(mark(X)) -> mark(incr(X))
         hd(mark(X)) -> mark(hd(X))
         tl(mark(X)) -> mark(tl(X))
         proper(nats()) -> ok(nats())
         proper(adx(X)) -> adx(proper(X))
         proper(zeros()) -> ok(zeros())
         proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
         proper(0()) -> ok(0())
         proper(incr(X)) -> incr(proper(X))
         proper(s(X)) -> s(proper(X))
         proper(hd(X)) -> hd(proper(X))
         proper(tl(X)) -> tl(proper(X))
         adx(ok(X)) -> ok(adx(X))
         cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
         incr(ok(X)) -> ok(incr(X))
         s(ok(X)) -> ok(s(X))
         hd(ok(X)) -> ok(hd(X))
         tl(ok(X)) -> ok(tl(X))
         top(mark(X)) -> top(proper(X))
         top(ok(X)) -> top(active(X))
        Usable Rule Processor:
         DPs:
          top#(ok(X)) -> top#(active(X))
         TRS:
          active(zeros()) -> mark(cons(0(),zeros()))
          active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
          active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
          active(hd(cons(X,Y))) -> mark(X)
          active(tl(cons(X,Y))) -> mark(Y)
          active(adx(X)) -> adx(active(X))
          active(incr(X)) -> incr(active(X))
          active(hd(X)) -> hd(active(X))
          active(tl(X)) -> tl(active(X))
          incr(mark(X)) -> mark(incr(X))
          incr(ok(X)) -> ok(incr(X))
          s(ok(X)) -> ok(s(X))
          cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
          adx(mark(X)) -> mark(adx(X))
          adx(ok(X)) -> ok(adx(X))
          hd(mark(X)) -> mark(hd(X))
          hd(ok(X)) -> ok(hd(X))
          tl(mark(X)) -> mark(tl(X))
          tl(ok(X)) -> ok(tl(X))
         Semantic Labeling Processor:
          dimension: 1
          usable rules:
           active(zeros()) -> mark(cons(0(),zeros()))
           active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
           active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
           active(hd(cons(X,Y))) -> mark(X)
           active(tl(cons(X,Y))) -> mark(Y)
           active(adx(X)) -> adx(active(X))
           active(incr(X)) -> incr(active(X))
           active(hd(X)) -> hd(active(X))
           active(tl(X)) -> tl(active(X))
           incr(mark(X)) -> mark(incr(X))
           incr(ok(X)) -> ok(incr(X))
           adx(mark(X)) -> mark(adx(X))
           adx(ok(X)) -> ok(adx(X))
           hd(mark(X)) -> mark(hd(X))
           hd(ok(X)) -> ok(hd(X))
           tl(mark(X)) -> mark(tl(X))
           tl(ok(X)) -> ok(tl(X))
          interpretation:
           [ok](x0) = 1,
           
           [tl](x0) = x0,
           
           [hd](x0) = 0,
           
           [s](x0) = x0,
           
           [incr](x0) = x0 + 1,
           
           [cons](x0, x1) = x0 + 1,
           
           [0] = 1,
           
           [mark](x0) = 1,
           
           [adx](x0) = 0,
           
           [zeros] = 0,
           
           [active](x0) = 0
           labeled:
           usable (for model):
           argument filtering:
            pi(active) = 0
            pi(zeros) = []
            pi(adx) = [0]
            pi(mark) = []
            pi(0) = []
            pi(cons) = []
            pi(incr) = [0]
            pi(s) = []
            pi(hd) = [0]
            pi(tl) = [0]
            pi(ok) = [0]
            pi(top#) = 0
          precedence:
           tl > hd > incr ~ adx ~ zeros > top# ~ ok ~ s ~ cons ~ 0 ~ mark ~ active
          problem:
           DPs:
            
           TRS:
            active(zeros()) -> mark(cons(0(),zeros()))
            active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
            active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
            active(hd(cons(X,Y))) -> mark(X)
            active(tl(cons(X,Y))) -> mark(Y)
            active(adx(X)) -> adx(active(X))
            active(incr(X)) -> incr(active(X))
            active(hd(X)) -> hd(active(X))
            active(tl(X)) -> tl(active(X))
            incr(mark(X)) -> mark(incr(X))
            incr(ok(X)) -> ok(incr(X))
            s(ok(X)) -> ok(s(X))
            cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
            adx(mark(X)) -> mark(adx(X))
            adx(ok(X)) -> ok(adx(X))
            hd(mark(X)) -> mark(hd(X))
            hd(ok(X)) -> ok(hd(X))
            tl(mark(X)) -> mark(tl(X))
            tl(ok(X)) -> ok(tl(X))
          Qed
     
     DPs:
      active#(adx(X)) -> active#(X)
      active#(incr(X)) -> active#(X)
      active#(hd(X)) -> active#(X)
      active#(tl(X)) -> active#(X)
     TRS:
      active(zeros()) -> mark(cons(0(),zeros()))
      active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
      active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
      active(hd(cons(X,Y))) -> mark(X)
      active(tl(cons(X,Y))) -> mark(Y)
      active(adx(X)) -> adx(active(X))
      active(incr(X)) -> incr(active(X))
      active(hd(X)) -> hd(active(X))
      active(tl(X)) -> tl(active(X))
      adx(mark(X)) -> mark(adx(X))
      incr(mark(X)) -> mark(incr(X))
      hd(mark(X)) -> mark(hd(X))
      tl(mark(X)) -> mark(tl(X))
      proper(nats()) -> ok(nats())
      proper(adx(X)) -> adx(proper(X))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(hd(X)) -> hd(proper(X))
      proper(tl(X)) -> tl(proper(X))
      adx(ok(X)) -> ok(adx(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      hd(ok(X)) -> ok(hd(X))
      tl(ok(X)) -> ok(tl(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(active#) = 0
      problem:
       DPs:
        
       TRS:
        active(zeros()) -> mark(cons(0(),zeros()))
        active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
        active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
        active(hd(cons(X,Y))) -> mark(X)
        active(tl(cons(X,Y))) -> mark(Y)
        active(adx(X)) -> adx(active(X))
        active(incr(X)) -> incr(active(X))
        active(hd(X)) -> hd(active(X))
        active(tl(X)) -> tl(active(X))
        adx(mark(X)) -> mark(adx(X))
        incr(mark(X)) -> mark(incr(X))
        hd(mark(X)) -> mark(hd(X))
        tl(mark(X)) -> mark(tl(X))
        proper(nats()) -> ok(nats())
        proper(adx(X)) -> adx(proper(X))
        proper(zeros()) -> ok(zeros())
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        proper(incr(X)) -> incr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(hd(X)) -> hd(proper(X))
        proper(tl(X)) -> tl(proper(X))
        adx(ok(X)) -> ok(adx(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        incr(ok(X)) -> ok(incr(X))
        s(ok(X)) -> ok(s(X))
        hd(ok(X)) -> ok(hd(X))
        tl(ok(X)) -> ok(tl(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
     
     DPs:
      proper#(adx(X)) -> proper#(X)
      proper#(cons(X1,X2)) -> proper#(X2)
      proper#(cons(X1,X2)) -> proper#(X1)
      proper#(incr(X)) -> proper#(X)
      proper#(s(X)) -> proper#(X)
      proper#(hd(X)) -> proper#(X)
      proper#(tl(X)) -> proper#(X)
     TRS:
      active(zeros()) -> mark(cons(0(),zeros()))
      active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
      active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
      active(hd(cons(X,Y))) -> mark(X)
      active(tl(cons(X,Y))) -> mark(Y)
      active(adx(X)) -> adx(active(X))
      active(incr(X)) -> incr(active(X))
      active(hd(X)) -> hd(active(X))
      active(tl(X)) -> tl(active(X))
      adx(mark(X)) -> mark(adx(X))
      incr(mark(X)) -> mark(incr(X))
      hd(mark(X)) -> mark(hd(X))
      tl(mark(X)) -> mark(tl(X))
      proper(nats()) -> ok(nats())
      proper(adx(X)) -> adx(proper(X))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(hd(X)) -> hd(proper(X))
      proper(tl(X)) -> tl(proper(X))
      adx(ok(X)) -> ok(adx(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      hd(ok(X)) -> ok(hd(X))
      tl(ok(X)) -> ok(tl(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(proper#) = 0
      problem:
       DPs:
        
       TRS:
        active(zeros()) -> mark(cons(0(),zeros()))
        active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
        active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
        active(hd(cons(X,Y))) -> mark(X)
        active(tl(cons(X,Y))) -> mark(Y)
        active(adx(X)) -> adx(active(X))
        active(incr(X)) -> incr(active(X))
        active(hd(X)) -> hd(active(X))
        active(tl(X)) -> tl(active(X))
        adx(mark(X)) -> mark(adx(X))
        incr(mark(X)) -> mark(incr(X))
        hd(mark(X)) -> mark(hd(X))
        tl(mark(X)) -> mark(tl(X))
        proper(nats()) -> ok(nats())
        proper(adx(X)) -> adx(proper(X))
        proper(zeros()) -> ok(zeros())
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        proper(incr(X)) -> incr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(hd(X)) -> hd(proper(X))
        proper(tl(X)) -> tl(proper(X))
        adx(ok(X)) -> ok(adx(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        incr(ok(X)) -> ok(incr(X))
        s(ok(X)) -> ok(s(X))
        hd(ok(X)) -> ok(hd(X))
        tl(ok(X)) -> ok(tl(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
     
     DPs:
      tl#(mark(X)) -> tl#(X)
      tl#(ok(X)) -> tl#(X)
     TRS:
      active(zeros()) -> mark(cons(0(),zeros()))
      active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
      active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
      active(hd(cons(X,Y))) -> mark(X)
      active(tl(cons(X,Y))) -> mark(Y)
      active(adx(X)) -> adx(active(X))
      active(incr(X)) -> incr(active(X))
      active(hd(X)) -> hd(active(X))
      active(tl(X)) -> tl(active(X))
      adx(mark(X)) -> mark(adx(X))
      incr(mark(X)) -> mark(incr(X))
      hd(mark(X)) -> mark(hd(X))
      tl(mark(X)) -> mark(tl(X))
      proper(nats()) -> ok(nats())
      proper(adx(X)) -> adx(proper(X))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(hd(X)) -> hd(proper(X))
      proper(tl(X)) -> tl(proper(X))
      adx(ok(X)) -> ok(adx(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      hd(ok(X)) -> ok(hd(X))
      tl(ok(X)) -> ok(tl(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(tl#) = 0
      problem:
       DPs:
        
       TRS:
        active(zeros()) -> mark(cons(0(),zeros()))
        active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
        active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
        active(hd(cons(X,Y))) -> mark(X)
        active(tl(cons(X,Y))) -> mark(Y)
        active(adx(X)) -> adx(active(X))
        active(incr(X)) -> incr(active(X))
        active(hd(X)) -> hd(active(X))
        active(tl(X)) -> tl(active(X))
        adx(mark(X)) -> mark(adx(X))
        incr(mark(X)) -> mark(incr(X))
        hd(mark(X)) -> mark(hd(X))
        tl(mark(X)) -> mark(tl(X))
        proper(nats()) -> ok(nats())
        proper(adx(X)) -> adx(proper(X))
        proper(zeros()) -> ok(zeros())
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        proper(incr(X)) -> incr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(hd(X)) -> hd(proper(X))
        proper(tl(X)) -> tl(proper(X))
        adx(ok(X)) -> ok(adx(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        incr(ok(X)) -> ok(incr(X))
        s(ok(X)) -> ok(s(X))
        hd(ok(X)) -> ok(hd(X))
        tl(ok(X)) -> ok(tl(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
     
     DPs:
      hd#(mark(X)) -> hd#(X)
      hd#(ok(X)) -> hd#(X)
     TRS:
      active(zeros()) -> mark(cons(0(),zeros()))
      active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
      active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
      active(hd(cons(X,Y))) -> mark(X)
      active(tl(cons(X,Y))) -> mark(Y)
      active(adx(X)) -> adx(active(X))
      active(incr(X)) -> incr(active(X))
      active(hd(X)) -> hd(active(X))
      active(tl(X)) -> tl(active(X))
      adx(mark(X)) -> mark(adx(X))
      incr(mark(X)) -> mark(incr(X))
      hd(mark(X)) -> mark(hd(X))
      tl(mark(X)) -> mark(tl(X))
      proper(nats()) -> ok(nats())
      proper(adx(X)) -> adx(proper(X))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(hd(X)) -> hd(proper(X))
      proper(tl(X)) -> tl(proper(X))
      adx(ok(X)) -> ok(adx(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      hd(ok(X)) -> ok(hd(X))
      tl(ok(X)) -> ok(tl(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(hd#) = 0
      problem:
       DPs:
        
       TRS:
        active(zeros()) -> mark(cons(0(),zeros()))
        active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
        active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
        active(hd(cons(X,Y))) -> mark(X)
        active(tl(cons(X,Y))) -> mark(Y)
        active(adx(X)) -> adx(active(X))
        active(incr(X)) -> incr(active(X))
        active(hd(X)) -> hd(active(X))
        active(tl(X)) -> tl(active(X))
        adx(mark(X)) -> mark(adx(X))
        incr(mark(X)) -> mark(incr(X))
        hd(mark(X)) -> mark(hd(X))
        tl(mark(X)) -> mark(tl(X))
        proper(nats()) -> ok(nats())
        proper(adx(X)) -> adx(proper(X))
        proper(zeros()) -> ok(zeros())
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        proper(incr(X)) -> incr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(hd(X)) -> hd(proper(X))
        proper(tl(X)) -> tl(proper(X))
        adx(ok(X)) -> ok(adx(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        incr(ok(X)) -> ok(incr(X))
        s(ok(X)) -> ok(s(X))
        hd(ok(X)) -> ok(hd(X))
        tl(ok(X)) -> ok(tl(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
     
     DPs:
      s#(ok(X)) -> s#(X)
     TRS:
      active(zeros()) -> mark(cons(0(),zeros()))
      active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
      active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
      active(hd(cons(X,Y))) -> mark(X)
      active(tl(cons(X,Y))) -> mark(Y)
      active(adx(X)) -> adx(active(X))
      active(incr(X)) -> incr(active(X))
      active(hd(X)) -> hd(active(X))
      active(tl(X)) -> tl(active(X))
      adx(mark(X)) -> mark(adx(X))
      incr(mark(X)) -> mark(incr(X))
      hd(mark(X)) -> mark(hd(X))
      tl(mark(X)) -> mark(tl(X))
      proper(nats()) -> ok(nats())
      proper(adx(X)) -> adx(proper(X))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(hd(X)) -> hd(proper(X))
      proper(tl(X)) -> tl(proper(X))
      adx(ok(X)) -> ok(adx(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      hd(ok(X)) -> ok(hd(X))
      tl(ok(X)) -> ok(tl(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(s#) = 0
      problem:
       DPs:
        
       TRS:
        active(zeros()) -> mark(cons(0(),zeros()))
        active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
        active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
        active(hd(cons(X,Y))) -> mark(X)
        active(tl(cons(X,Y))) -> mark(Y)
        active(adx(X)) -> adx(active(X))
        active(incr(X)) -> incr(active(X))
        active(hd(X)) -> hd(active(X))
        active(tl(X)) -> tl(active(X))
        adx(mark(X)) -> mark(adx(X))
        incr(mark(X)) -> mark(incr(X))
        hd(mark(X)) -> mark(hd(X))
        tl(mark(X)) -> mark(tl(X))
        proper(nats()) -> ok(nats())
        proper(adx(X)) -> adx(proper(X))
        proper(zeros()) -> ok(zeros())
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        proper(incr(X)) -> incr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(hd(X)) -> hd(proper(X))
        proper(tl(X)) -> tl(proper(X))
        adx(ok(X)) -> ok(adx(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        incr(ok(X)) -> ok(incr(X))
        s(ok(X)) -> ok(s(X))
        hd(ok(X)) -> ok(hd(X))
        tl(ok(X)) -> ok(tl(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
     
     DPs:
      incr#(mark(X)) -> incr#(X)
      incr#(ok(X)) -> incr#(X)
     TRS:
      active(zeros()) -> mark(cons(0(),zeros()))
      active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
      active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
      active(hd(cons(X,Y))) -> mark(X)
      active(tl(cons(X,Y))) -> mark(Y)
      active(adx(X)) -> adx(active(X))
      active(incr(X)) -> incr(active(X))
      active(hd(X)) -> hd(active(X))
      active(tl(X)) -> tl(active(X))
      adx(mark(X)) -> mark(adx(X))
      incr(mark(X)) -> mark(incr(X))
      hd(mark(X)) -> mark(hd(X))
      tl(mark(X)) -> mark(tl(X))
      proper(nats()) -> ok(nats())
      proper(adx(X)) -> adx(proper(X))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(hd(X)) -> hd(proper(X))
      proper(tl(X)) -> tl(proper(X))
      adx(ok(X)) -> ok(adx(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      hd(ok(X)) -> ok(hd(X))
      tl(ok(X)) -> ok(tl(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(incr#) = 0
      problem:
       DPs:
        
       TRS:
        active(zeros()) -> mark(cons(0(),zeros()))
        active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
        active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
        active(hd(cons(X,Y))) -> mark(X)
        active(tl(cons(X,Y))) -> mark(Y)
        active(adx(X)) -> adx(active(X))
        active(incr(X)) -> incr(active(X))
        active(hd(X)) -> hd(active(X))
        active(tl(X)) -> tl(active(X))
        adx(mark(X)) -> mark(adx(X))
        incr(mark(X)) -> mark(incr(X))
        hd(mark(X)) -> mark(hd(X))
        tl(mark(X)) -> mark(tl(X))
        proper(nats()) -> ok(nats())
        proper(adx(X)) -> adx(proper(X))
        proper(zeros()) -> ok(zeros())
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        proper(incr(X)) -> incr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(hd(X)) -> hd(proper(X))
        proper(tl(X)) -> tl(proper(X))
        adx(ok(X)) -> ok(adx(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        incr(ok(X)) -> ok(incr(X))
        s(ok(X)) -> ok(s(X))
        hd(ok(X)) -> ok(hd(X))
        tl(ok(X)) -> ok(tl(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
     
     DPs:
      cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     TRS:
      active(zeros()) -> mark(cons(0(),zeros()))
      active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
      active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
      active(hd(cons(X,Y))) -> mark(X)
      active(tl(cons(X,Y))) -> mark(Y)
      active(adx(X)) -> adx(active(X))
      active(incr(X)) -> incr(active(X))
      active(hd(X)) -> hd(active(X))
      active(tl(X)) -> tl(active(X))
      adx(mark(X)) -> mark(adx(X))
      incr(mark(X)) -> mark(incr(X))
      hd(mark(X)) -> mark(hd(X))
      tl(mark(X)) -> mark(tl(X))
      proper(nats()) -> ok(nats())
      proper(adx(X)) -> adx(proper(X))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(hd(X)) -> hd(proper(X))
      proper(tl(X)) -> tl(proper(X))
      adx(ok(X)) -> ok(adx(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      hd(ok(X)) -> ok(hd(X))
      tl(ok(X)) -> ok(tl(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(cons#) = 1
      problem:
       DPs:
        
       TRS:
        active(zeros()) -> mark(cons(0(),zeros()))
        active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
        active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
        active(hd(cons(X,Y))) -> mark(X)
        active(tl(cons(X,Y))) -> mark(Y)
        active(adx(X)) -> adx(active(X))
        active(incr(X)) -> incr(active(X))
        active(hd(X)) -> hd(active(X))
        active(tl(X)) -> tl(active(X))
        adx(mark(X)) -> mark(adx(X))
        incr(mark(X)) -> mark(incr(X))
        hd(mark(X)) -> mark(hd(X))
        tl(mark(X)) -> mark(tl(X))
        proper(nats()) -> ok(nats())
        proper(adx(X)) -> adx(proper(X))
        proper(zeros()) -> ok(zeros())
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        proper(incr(X)) -> incr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(hd(X)) -> hd(proper(X))
        proper(tl(X)) -> tl(proper(X))
        adx(ok(X)) -> ok(adx(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        incr(ok(X)) -> ok(incr(X))
        s(ok(X)) -> ok(s(X))
        hd(ok(X)) -> ok(hd(X))
        tl(ok(X)) -> ok(tl(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
     
     DPs:
      adx#(mark(X)) -> adx#(X)
      adx#(ok(X)) -> adx#(X)
     TRS:
      active(zeros()) -> mark(cons(0(),zeros()))
      active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
      active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
      active(hd(cons(X,Y))) -> mark(X)
      active(tl(cons(X,Y))) -> mark(Y)
      active(adx(X)) -> adx(active(X))
      active(incr(X)) -> incr(active(X))
      active(hd(X)) -> hd(active(X))
      active(tl(X)) -> tl(active(X))
      adx(mark(X)) -> mark(adx(X))
      incr(mark(X)) -> mark(incr(X))
      hd(mark(X)) -> mark(hd(X))
      tl(mark(X)) -> mark(tl(X))
      proper(nats()) -> ok(nats())
      proper(adx(X)) -> adx(proper(X))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(hd(X)) -> hd(proper(X))
      proper(tl(X)) -> tl(proper(X))
      adx(ok(X)) -> ok(adx(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      hd(ok(X)) -> ok(hd(X))
      tl(ok(X)) -> ok(tl(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(adx#) = 0
      problem:
       DPs:
        
       TRS:
        active(zeros()) -> mark(cons(0(),zeros()))
        active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y)))
        active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y))))
        active(hd(cons(X,Y))) -> mark(X)
        active(tl(cons(X,Y))) -> mark(Y)
        active(adx(X)) -> adx(active(X))
        active(incr(X)) -> incr(active(X))
        active(hd(X)) -> hd(active(X))
        active(tl(X)) -> tl(active(X))
        adx(mark(X)) -> mark(adx(X))
        incr(mark(X)) -> mark(incr(X))
        hd(mark(X)) -> mark(hd(X))
        tl(mark(X)) -> mark(tl(X))
        proper(nats()) -> ok(nats())
        proper(adx(X)) -> adx(proper(X))
        proper(zeros()) -> ok(zeros())
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        proper(incr(X)) -> incr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(hd(X)) -> hd(proper(X))
        proper(tl(X)) -> tl(proper(X))
        adx(ok(X)) -> ok(adx(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        incr(ok(X)) -> ok(incr(X))
        s(ok(X)) -> ok(s(X))
        hd(ok(X)) -> ok(hd(X))
        tl(ok(X)) -> ok(tl(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed