Problem:
 active(zeros()) -> mark(cons(0(),zeros()))
 active(and(tt(),X)) -> mark(X)
 active(length(nil())) -> mark(0())
 active(length(cons(N,L))) -> mark(s(length(L)))
 active(take(0(),IL)) -> mark(nil())
 active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL)))
 mark(zeros()) -> active(zeros())
 mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
 mark(0()) -> active(0())
 mark(and(X1,X2)) -> active(and(mark(X1),X2))
 mark(tt()) -> active(tt())
 mark(length(X)) -> active(length(mark(X)))
 mark(nil()) -> active(nil())
 mark(s(X)) -> active(s(mark(X)))
 mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
 cons(active(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 and(mark(X1),X2) -> and(X1,X2)
 and(X1,mark(X2)) -> and(X1,X2)
 and(active(X1),X2) -> and(X1,X2)
 and(X1,active(X2)) -> and(X1,X2)
 length(mark(X)) -> length(X)
 length(active(X)) -> length(X)
 s(mark(X)) -> s(X)
 s(active(X)) -> s(X)
 take(mark(X1),X2) -> take(X1,X2)
 take(X1,mark(X2)) -> take(X1,X2)
 take(active(X1),X2) -> take(X1,X2)
 take(X1,active(X2)) -> take(X1,X2)

Proof:
 Bounds Processor:
  bound: 4
  enrichment: match
  automaton:
   final states: {11,10,9,8,7,6,5}
   transitions:
    active1(15) -> 6*
    active1(16) -> 6*
    nil1() -> 15*
    tt1() -> 15*
    01() -> 16*
    zeros1() -> 15*
    mark1(17) -> 5*
    cons1(16,15) -> 17*
    active2(27) -> 5*
    active2(19) -> 26*
    cons2(19,18) -> 20*
    cons2(26,15) -> 27*
    mark2(20) -> 6*
    mark2(16) -> 26*
    02() -> 19*
    zeros2() -> 18*
    active0(2) -> 5*
    active0(4) -> 5*
    active0(1) -> 5*
    active0(3) -> 5*
    cons3(28,18) -> 29*
    cons3(19,15) -> 27*
    cons3(16,15) -> 27*
    zeros0() -> 1*
    active3(29) -> 6*
    active3(33) -> 28*
    mark0(2) -> 6*
    mark0(4) -> 6*
    mark0(1) -> 6*
    mark0(3) -> 6*
    mark3(19) -> 28*
    cons0(3,1) -> 7*
    cons0(3,3) -> 7*
    cons0(4,2) -> 7*
    cons0(4,4) -> 7*
    cons0(1,2) -> 7*
    cons0(1,4) -> 7*
    cons0(2,1) -> 7*
    cons0(2,3) -> 7*
    cons0(3,2) -> 7*
    cons0(3,4) -> 7*
    cons0(4,1) -> 7*
    cons0(4,3) -> 7*
    cons0(1,1) -> 7*
    cons0(1,3) -> 7*
    cons0(2,2) -> 7*
    cons0(2,4) -> 7*
    cons4(19,18) -> 29*
    cons4(33,18) -> 29*
    00() -> 2*
    03() -> 33*
    and0(3,1) -> 8*
    and0(3,3) -> 8*
    and0(4,2) -> 8*
    and0(4,4) -> 8*
    and0(1,2) -> 8*
    and0(1,4) -> 8*
    and0(2,1) -> 8*
    and0(2,3) -> 8*
    and0(3,2) -> 8*
    and0(3,4) -> 8*
    and0(4,1) -> 8*
    and0(4,3) -> 8*
    and0(1,1) -> 8*
    and0(1,3) -> 8*
    and0(2,2) -> 8*
    and0(2,4) -> 8*
    tt0() -> 3*
    length0(2) -> 9*
    length0(4) -> 9*
    length0(1) -> 9*
    length0(3) -> 9*
    nil0() -> 4*
    s0(2) -> 10*
    s0(4) -> 10*
    s0(1) -> 10*
    s0(3) -> 10*
    take0(3,1) -> 11*
    take0(3,3) -> 11*
    take0(4,2) -> 11*
    take0(4,4) -> 11*
    take0(1,2) -> 11*
    take0(1,4) -> 11*
    take0(2,1) -> 11*
    take0(2,3) -> 11*
    take0(3,2) -> 11*
    take0(3,4) -> 11*
    take0(4,1) -> 11*
    take0(4,3) -> 11*
    take0(1,1) -> 11*
    take0(1,3) -> 11*
    take0(2,2) -> 11*
    take0(2,4) -> 11*
  problem:
   
  Qed