WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxRelTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 38 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 1405 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 2807 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   A__ZEROS -> c
   A__ZEROS -> c1
   A__AND(tt, z0) -> c2(MARK(z0))
   A__AND(z0, z1) -> c3
   A__LENGTH(nil) -> c4
   A__LENGTH(cons(z0, z1)) -> c5(A__LENGTH(mark(z1)), MARK(z1))
   A__LENGTH(z0) -> c6
   A__TAKE(0, z0) -> c7
   A__TAKE(s(z0), cons(z1, z2)) -> c8(MARK(z1))
   A__TAKE(z0, z1) -> c9
   MARK(zeros) -> c10(A__ZEROS)
   MARK(and(z0, z1)) -> c11(A__AND(mark(z0), z1), MARK(z0))
   MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c13(A__TAKE(mark(z0), mark(z1)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z1))
   MARK(cons(z0, z1)) -> c15(MARK(z0))
   MARK(0) -> c16
   MARK(tt) -> c17
   MARK(nil) -> c18
   MARK(s(z0)) -> c19(MARK(z0))

The (relative) TRS S consists of the following rules:

   a__zeros -> cons(0, zeros)
   a__zeros -> zeros
   a__and(tt, z0) -> mark(z0)
   a__and(z0, z1) -> and(z0, z1)
   a__length(nil) -> 0
   a__length(cons(z0, z1)) -> s(a__length(mark(z1)))
   a__length(z0) -> length(z0)
   a__take(0, z0) -> nil
   a__take(s(z0), cons(z1, z2)) -> cons(mark(z1), take(z0, z2))
   a__take(z0, z1) -> take(z0, z1)
   mark(zeros) -> a__zeros
   mark(and(z0, z1)) -> a__and(mark(z0), z1)
   mark(length(z0)) -> a__length(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(0) -> 0
   mark(tt) -> tt
   mark(nil) -> nil
   mark(s(z0)) -> s(mark(z0))

Rewrite Strategy: INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   A__ZEROS -> c
   A__ZEROS -> c1
   A__AND(tt, z0) -> c2(MARK(z0))
   A__AND(z0, z1) -> c3
   A__LENGTH(nil) -> c4
   A__LENGTH(cons(z0, z1)) -> c5(A__LENGTH(mark(z1)), MARK(z1))
   A__LENGTH(z0) -> c6
   A__TAKE(0', z0) -> c7
   A__TAKE(s(z0), cons(z1, z2)) -> c8(MARK(z1))
   A__TAKE(z0, z1) -> c9
   MARK(zeros) -> c10(A__ZEROS)
   MARK(and(z0, z1)) -> c11(A__AND(mark(z0), z1), MARK(z0))
   MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c13(A__TAKE(mark(z0), mark(z1)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z1))
   MARK(cons(z0, z1)) -> c15(MARK(z0))
   MARK(0') -> c16
   MARK(tt) -> c17
   MARK(nil) -> c18
   MARK(s(z0)) -> c19(MARK(z0))

The (relative) TRS S consists of the following rules:

   a__zeros -> cons(0', zeros)
   a__zeros -> zeros
   a__and(tt, z0) -> mark(z0)
   a__and(z0, z1) -> and(z0, z1)
   a__length(nil) -> 0'
   a__length(cons(z0, z1)) -> s(a__length(mark(z1)))
   a__length(z0) -> length(z0)
   a__take(0', z0) -> nil
   a__take(s(z0), cons(z1, z2)) -> cons(mark(z1), take(z0, z2))
   a__take(z0, z1) -> take(z0, z1)
   mark(zeros) -> a__zeros
   mark(and(z0, z1)) -> a__and(mark(z0), z1)
   mark(length(z0)) -> a__length(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(0') -> 0'
   mark(tt) -> tt
   mark(nil) -> nil
   mark(s(z0)) -> s(mark(z0))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Innermost TRS:
Rules:
A__ZEROS -> c
A__ZEROS -> c1
A__AND(tt, z0) -> c2(MARK(z0))
A__AND(z0, z1) -> c3
A__LENGTH(nil) -> c4
A__LENGTH(cons(z0, z1)) -> c5(A__LENGTH(mark(z1)), MARK(z1))
A__LENGTH(z0) -> c6
A__TAKE(0', z0) -> c7
A__TAKE(s(z0), cons(z1, z2)) -> c8(MARK(z1))
A__TAKE(z0, z1) -> c9
MARK(zeros) -> c10(A__ZEROS)
MARK(and(z0, z1)) -> c11(A__AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0))
MARK(take(z0, z1)) -> c13(A__TAKE(mark(z0), mark(z1)), MARK(z0))
MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z1))
MARK(cons(z0, z1)) -> c15(MARK(z0))
MARK(0') -> c16
MARK(tt) -> c17
MARK(nil) -> c18
MARK(s(z0)) -> c19(MARK(z0))
a__zeros -> cons(0', zeros)
a__zeros -> zeros
a__and(tt, z0) -> mark(z0)
a__and(z0, z1) -> and(z0, z1)
a__length(nil) -> 0'
a__length(cons(z0, z1)) -> s(a__length(mark(z1)))
a__length(z0) -> length(z0)
a__take(0', z0) -> nil
a__take(s(z0), cons(z1, z2)) -> cons(mark(z1), take(z0, z2))
a__take(z0, z1) -> take(z0, z1)
mark(zeros) -> a__zeros
mark(and(z0, z1)) -> a__and(mark(z0), z1)
mark(length(z0)) -> a__length(mark(z0))
mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
mark(cons(z0, z1)) -> cons(mark(z0), z1)
mark(0') -> 0'
mark(tt) -> tt
mark(nil) -> nil
mark(s(z0)) -> s(mark(z0))

Types:
A__ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
A__AND :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c2:c3
tt :: tt:nil:cons:0':s:zeros:and:length:take
c2 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c2:c3
MARK :: tt:nil:cons:0':s:zeros:and:length:take -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c3 :: c2:c3
A__LENGTH :: tt:nil:cons:0':s:zeros:and:length:take -> c4:c5:c6
nil :: tt:nil:cons:0':s:zeros:and:length:take
c4 :: c4:c5:c6
cons :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c5 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c4:c5:c6
mark :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c6 :: c4:c5:c6
A__TAKE :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c7:c8:c9
0' :: tt:nil:cons:0':s:zeros:and:length:take
c7 :: c7:c8:c9
s :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c8 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c7:c8:c9
c9 :: c7:c8:c9
zeros :: tt:nil:cons:0':s:zeros:and:length:take
c10 :: c:c1 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c11 :: c2:c3 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c12 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c13 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c14 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c15 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c16 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c17 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c18 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c19 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
a__zeros :: tt:nil:cons:0':s:zeros:and:length:take
a__and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
hole_c:c11_20 :: c:c1
hole_c2:c32_20 :: c2:c3
hole_tt:nil:cons:0':s:zeros:and:length:take3_20 :: tt:nil:cons:0':s:zeros:and:length:take
hole_c10:c11:c12:c13:c14:c15:c16:c17:c18:c194_20 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
hole_c4:c5:c65_20 :: c4:c5:c6
hole_c7:c8:c96_20 :: c7:c8:c9
gen_tt:nil:cons:0':s:zeros:and:length:take7_20 :: Nat -> tt:nil:cons:0':s:zeros:and:length:take
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20 :: Nat -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
gen_c4:c5:c69_20 :: Nat -> c4:c5:c6

----------------------------------------

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MARK, A__LENGTH, mark, a__length

They will be analysed ascendingly in the following order:
MARK = A__LENGTH
mark < MARK
mark < A__LENGTH
mark = a__length

----------------------------------------

(6)
Obligation:
Innermost TRS:
Rules:
A__ZEROS -> c
A__ZEROS -> c1
A__AND(tt, z0) -> c2(MARK(z0))
A__AND(z0, z1) -> c3
A__LENGTH(nil) -> c4
A__LENGTH(cons(z0, z1)) -> c5(A__LENGTH(mark(z1)), MARK(z1))
A__LENGTH(z0) -> c6
A__TAKE(0', z0) -> c7
A__TAKE(s(z0), cons(z1, z2)) -> c8(MARK(z1))
A__TAKE(z0, z1) -> c9
MARK(zeros) -> c10(A__ZEROS)
MARK(and(z0, z1)) -> c11(A__AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0))
MARK(take(z0, z1)) -> c13(A__TAKE(mark(z0), mark(z1)), MARK(z0))
MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z1))
MARK(cons(z0, z1)) -> c15(MARK(z0))
MARK(0') -> c16
MARK(tt) -> c17
MARK(nil) -> c18
MARK(s(z0)) -> c19(MARK(z0))
a__zeros -> cons(0', zeros)
a__zeros -> zeros
a__and(tt, z0) -> mark(z0)
a__and(z0, z1) -> and(z0, z1)
a__length(nil) -> 0'
a__length(cons(z0, z1)) -> s(a__length(mark(z1)))
a__length(z0) -> length(z0)
a__take(0', z0) -> nil
a__take(s(z0), cons(z1, z2)) -> cons(mark(z1), take(z0, z2))
a__take(z0, z1) -> take(z0, z1)
mark(zeros) -> a__zeros
mark(and(z0, z1)) -> a__and(mark(z0), z1)
mark(length(z0)) -> a__length(mark(z0))
mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
mark(cons(z0, z1)) -> cons(mark(z0), z1)
mark(0') -> 0'
mark(tt) -> tt
mark(nil) -> nil
mark(s(z0)) -> s(mark(z0))

Types:
A__ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
A__AND :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c2:c3
tt :: tt:nil:cons:0':s:zeros:and:length:take
c2 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c2:c3
MARK :: tt:nil:cons:0':s:zeros:and:length:take -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c3 :: c2:c3
A__LENGTH :: tt:nil:cons:0':s:zeros:and:length:take -> c4:c5:c6
nil :: tt:nil:cons:0':s:zeros:and:length:take
c4 :: c4:c5:c6
cons :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c5 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c4:c5:c6
mark :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c6 :: c4:c5:c6
A__TAKE :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c7:c8:c9
0' :: tt:nil:cons:0':s:zeros:and:length:take
c7 :: c7:c8:c9
s :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c8 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c7:c8:c9
c9 :: c7:c8:c9
zeros :: tt:nil:cons:0':s:zeros:and:length:take
c10 :: c:c1 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c11 :: c2:c3 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c12 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c13 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c14 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c15 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c16 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c17 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c18 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c19 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
a__zeros :: tt:nil:cons:0':s:zeros:and:length:take
a__and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
hole_c:c11_20 :: c:c1
hole_c2:c32_20 :: c2:c3
hole_tt:nil:cons:0':s:zeros:and:length:take3_20 :: tt:nil:cons:0':s:zeros:and:length:take
hole_c10:c11:c12:c13:c14:c15:c16:c17:c18:c194_20 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
hole_c4:c5:c65_20 :: c4:c5:c6
hole_c7:c8:c96_20 :: c7:c8:c9
gen_tt:nil:cons:0':s:zeros:and:length:take7_20 :: Nat -> tt:nil:cons:0':s:zeros:and:length:take
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20 :: Nat -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
gen_c4:c5:c69_20 :: Nat -> c4:c5:c6


Generator Equations:
gen_tt:nil:cons:0':s:zeros:and:length:take7_20(0) <=> tt
gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(x, 1)) <=> cons(tt, gen_tt:nil:cons:0':s:zeros:and:length:take7_20(x))
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(0) <=> c10(c)
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(+(x, 1)) <=> c11(c2(c10(c)), gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(x))
gen_c4:c5:c69_20(0) <=> c4
gen_c4:c5:c69_20(+(x, 1)) <=> c5(gen_c4:c5:c69_20(x), c10(c))


The following defined symbols remain to be analysed:
a__length, MARK, A__LENGTH, mark

They will be analysed ascendingly in the following order:
MARK = A__LENGTH
mark < MARK
mark < A__LENGTH
mark = a__length

----------------------------------------

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a__length(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n11_20))) -> *10_20, rt in Omega(0)

Induction Base:
a__length(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, 0)))

Induction Step:
a__length(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, +(n11_20, 1)))) ->_R^Omega(0)
s(a__length(mark(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n11_20))))) ->_R^Omega(0)
s(a__length(cons(mark(tt), gen_tt:nil:cons:0':s:zeros:and:length:take7_20(n11_20)))) ->_R^Omega(0)
s(a__length(cons(tt, gen_tt:nil:cons:0':s:zeros:and:length:take7_20(n11_20)))) ->_IH
s(*10_20)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(8)
Obligation:
Innermost TRS:
Rules:
A__ZEROS -> c
A__ZEROS -> c1
A__AND(tt, z0) -> c2(MARK(z0))
A__AND(z0, z1) -> c3
A__LENGTH(nil) -> c4
A__LENGTH(cons(z0, z1)) -> c5(A__LENGTH(mark(z1)), MARK(z1))
A__LENGTH(z0) -> c6
A__TAKE(0', z0) -> c7
A__TAKE(s(z0), cons(z1, z2)) -> c8(MARK(z1))
A__TAKE(z0, z1) -> c9
MARK(zeros) -> c10(A__ZEROS)
MARK(and(z0, z1)) -> c11(A__AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0))
MARK(take(z0, z1)) -> c13(A__TAKE(mark(z0), mark(z1)), MARK(z0))
MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z1))
MARK(cons(z0, z1)) -> c15(MARK(z0))
MARK(0') -> c16
MARK(tt) -> c17
MARK(nil) -> c18
MARK(s(z0)) -> c19(MARK(z0))
a__zeros -> cons(0', zeros)
a__zeros -> zeros
a__and(tt, z0) -> mark(z0)
a__and(z0, z1) -> and(z0, z1)
a__length(nil) -> 0'
a__length(cons(z0, z1)) -> s(a__length(mark(z1)))
a__length(z0) -> length(z0)
a__take(0', z0) -> nil
a__take(s(z0), cons(z1, z2)) -> cons(mark(z1), take(z0, z2))
a__take(z0, z1) -> take(z0, z1)
mark(zeros) -> a__zeros
mark(and(z0, z1)) -> a__and(mark(z0), z1)
mark(length(z0)) -> a__length(mark(z0))
mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
mark(cons(z0, z1)) -> cons(mark(z0), z1)
mark(0') -> 0'
mark(tt) -> tt
mark(nil) -> nil
mark(s(z0)) -> s(mark(z0))

Types:
A__ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
A__AND :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c2:c3
tt :: tt:nil:cons:0':s:zeros:and:length:take
c2 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c2:c3
MARK :: tt:nil:cons:0':s:zeros:and:length:take -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c3 :: c2:c3
A__LENGTH :: tt:nil:cons:0':s:zeros:and:length:take -> c4:c5:c6
nil :: tt:nil:cons:0':s:zeros:and:length:take
c4 :: c4:c5:c6
cons :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c5 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c4:c5:c6
mark :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c6 :: c4:c5:c6
A__TAKE :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c7:c8:c9
0' :: tt:nil:cons:0':s:zeros:and:length:take
c7 :: c7:c8:c9
s :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c8 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c7:c8:c9
c9 :: c7:c8:c9
zeros :: tt:nil:cons:0':s:zeros:and:length:take
c10 :: c:c1 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c11 :: c2:c3 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c12 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c13 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c14 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c15 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c16 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c17 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c18 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c19 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
a__zeros :: tt:nil:cons:0':s:zeros:and:length:take
a__and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
hole_c:c11_20 :: c:c1
hole_c2:c32_20 :: c2:c3
hole_tt:nil:cons:0':s:zeros:and:length:take3_20 :: tt:nil:cons:0':s:zeros:and:length:take
hole_c10:c11:c12:c13:c14:c15:c16:c17:c18:c194_20 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
hole_c4:c5:c65_20 :: c4:c5:c6
hole_c7:c8:c96_20 :: c7:c8:c9
gen_tt:nil:cons:0':s:zeros:and:length:take7_20 :: Nat -> tt:nil:cons:0':s:zeros:and:length:take
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20 :: Nat -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
gen_c4:c5:c69_20 :: Nat -> c4:c5:c6


Lemmas:
a__length(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n11_20))) -> *10_20, rt in Omega(0)


Generator Equations:
gen_tt:nil:cons:0':s:zeros:and:length:take7_20(0) <=> tt
gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(x, 1)) <=> cons(tt, gen_tt:nil:cons:0':s:zeros:and:length:take7_20(x))
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(0) <=> c10(c)
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(+(x, 1)) <=> c11(c2(c10(c)), gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(x))
gen_c4:c5:c69_20(0) <=> c4
gen_c4:c5:c69_20(+(x, 1)) <=> c5(gen_c4:c5:c69_20(x), c10(c))


The following defined symbols remain to be analysed:
mark, MARK, A__LENGTH

They will be analysed ascendingly in the following order:
MARK = A__LENGTH
mark < MARK
mark < A__LENGTH
mark = a__length

----------------------------------------

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
A__LENGTH(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n39957_20))) -> *10_20, rt in Omega(n39957_20)

Induction Base:
A__LENGTH(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, 0)))

Induction Step:
A__LENGTH(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, +(n39957_20, 1)))) ->_R^Omega(1)
c5(A__LENGTH(mark(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n39957_20)))), MARK(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n39957_20)))) ->_R^Omega(0)
c5(A__LENGTH(cons(mark(tt), gen_tt:nil:cons:0':s:zeros:and:length:take7_20(n39957_20))), MARK(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n39957_20)))) ->_R^Omega(0)
c5(A__LENGTH(cons(tt, gen_tt:nil:cons:0':s:zeros:and:length:take7_20(n39957_20))), MARK(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n39957_20)))) ->_IH
c5(*10_20, MARK(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n39957_20)))) ->_R^Omega(1)
c5(*10_20, c15(MARK(tt))) ->_R^Omega(1)
c5(*10_20, c15(c17))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(10)
Complex Obligation (BEST)

----------------------------------------

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
A__ZEROS -> c
A__ZEROS -> c1
A__AND(tt, z0) -> c2(MARK(z0))
A__AND(z0, z1) -> c3
A__LENGTH(nil) -> c4
A__LENGTH(cons(z0, z1)) -> c5(A__LENGTH(mark(z1)), MARK(z1))
A__LENGTH(z0) -> c6
A__TAKE(0', z0) -> c7
A__TAKE(s(z0), cons(z1, z2)) -> c8(MARK(z1))
A__TAKE(z0, z1) -> c9
MARK(zeros) -> c10(A__ZEROS)
MARK(and(z0, z1)) -> c11(A__AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0))
MARK(take(z0, z1)) -> c13(A__TAKE(mark(z0), mark(z1)), MARK(z0))
MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z1))
MARK(cons(z0, z1)) -> c15(MARK(z0))
MARK(0') -> c16
MARK(tt) -> c17
MARK(nil) -> c18
MARK(s(z0)) -> c19(MARK(z0))
a__zeros -> cons(0', zeros)
a__zeros -> zeros
a__and(tt, z0) -> mark(z0)
a__and(z0, z1) -> and(z0, z1)
a__length(nil) -> 0'
a__length(cons(z0, z1)) -> s(a__length(mark(z1)))
a__length(z0) -> length(z0)
a__take(0', z0) -> nil
a__take(s(z0), cons(z1, z2)) -> cons(mark(z1), take(z0, z2))
a__take(z0, z1) -> take(z0, z1)
mark(zeros) -> a__zeros
mark(and(z0, z1)) -> a__and(mark(z0), z1)
mark(length(z0)) -> a__length(mark(z0))
mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
mark(cons(z0, z1)) -> cons(mark(z0), z1)
mark(0') -> 0'
mark(tt) -> tt
mark(nil) -> nil
mark(s(z0)) -> s(mark(z0))

Types:
A__ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
A__AND :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c2:c3
tt :: tt:nil:cons:0':s:zeros:and:length:take
c2 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c2:c3
MARK :: tt:nil:cons:0':s:zeros:and:length:take -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c3 :: c2:c3
A__LENGTH :: tt:nil:cons:0':s:zeros:and:length:take -> c4:c5:c6
nil :: tt:nil:cons:0':s:zeros:and:length:take
c4 :: c4:c5:c6
cons :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c5 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c4:c5:c6
mark :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c6 :: c4:c5:c6
A__TAKE :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c7:c8:c9
0' :: tt:nil:cons:0':s:zeros:and:length:take
c7 :: c7:c8:c9
s :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c8 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c7:c8:c9
c9 :: c7:c8:c9
zeros :: tt:nil:cons:0':s:zeros:and:length:take
c10 :: c:c1 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c11 :: c2:c3 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c12 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c13 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c14 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c15 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c16 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c17 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c18 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c19 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
a__zeros :: tt:nil:cons:0':s:zeros:and:length:take
a__and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
hole_c:c11_20 :: c:c1
hole_c2:c32_20 :: c2:c3
hole_tt:nil:cons:0':s:zeros:and:length:take3_20 :: tt:nil:cons:0':s:zeros:and:length:take
hole_c10:c11:c12:c13:c14:c15:c16:c17:c18:c194_20 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
hole_c4:c5:c65_20 :: c4:c5:c6
hole_c7:c8:c96_20 :: c7:c8:c9
gen_tt:nil:cons:0':s:zeros:and:length:take7_20 :: Nat -> tt:nil:cons:0':s:zeros:and:length:take
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20 :: Nat -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
gen_c4:c5:c69_20 :: Nat -> c4:c5:c6


Lemmas:
a__length(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n11_20))) -> *10_20, rt in Omega(0)


Generator Equations:
gen_tt:nil:cons:0':s:zeros:and:length:take7_20(0) <=> tt
gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(x, 1)) <=> cons(tt, gen_tt:nil:cons:0':s:zeros:and:length:take7_20(x))
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(0) <=> c10(c)
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(+(x, 1)) <=> c11(c2(c10(c)), gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(x))
gen_c4:c5:c69_20(0) <=> c4
gen_c4:c5:c69_20(+(x, 1)) <=> c5(gen_c4:c5:c69_20(x), c10(c))


The following defined symbols remain to be analysed:
A__LENGTH, MARK

They will be analysed ascendingly in the following order:
MARK = A__LENGTH

----------------------------------------

(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(13)
BOUNDS(n^1, INF)

----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
A__ZEROS -> c
A__ZEROS -> c1
A__AND(tt, z0) -> c2(MARK(z0))
A__AND(z0, z1) -> c3
A__LENGTH(nil) -> c4
A__LENGTH(cons(z0, z1)) -> c5(A__LENGTH(mark(z1)), MARK(z1))
A__LENGTH(z0) -> c6
A__TAKE(0', z0) -> c7
A__TAKE(s(z0), cons(z1, z2)) -> c8(MARK(z1))
A__TAKE(z0, z1) -> c9
MARK(zeros) -> c10(A__ZEROS)
MARK(and(z0, z1)) -> c11(A__AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0))
MARK(take(z0, z1)) -> c13(A__TAKE(mark(z0), mark(z1)), MARK(z0))
MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z1))
MARK(cons(z0, z1)) -> c15(MARK(z0))
MARK(0') -> c16
MARK(tt) -> c17
MARK(nil) -> c18
MARK(s(z0)) -> c19(MARK(z0))
a__zeros -> cons(0', zeros)
a__zeros -> zeros
a__and(tt, z0) -> mark(z0)
a__and(z0, z1) -> and(z0, z1)
a__length(nil) -> 0'
a__length(cons(z0, z1)) -> s(a__length(mark(z1)))
a__length(z0) -> length(z0)
a__take(0', z0) -> nil
a__take(s(z0), cons(z1, z2)) -> cons(mark(z1), take(z0, z2))
a__take(z0, z1) -> take(z0, z1)
mark(zeros) -> a__zeros
mark(and(z0, z1)) -> a__and(mark(z0), z1)
mark(length(z0)) -> a__length(mark(z0))
mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
mark(cons(z0, z1)) -> cons(mark(z0), z1)
mark(0') -> 0'
mark(tt) -> tt
mark(nil) -> nil
mark(s(z0)) -> s(mark(z0))

Types:
A__ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
A__AND :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c2:c3
tt :: tt:nil:cons:0':s:zeros:and:length:take
c2 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c2:c3
MARK :: tt:nil:cons:0':s:zeros:and:length:take -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c3 :: c2:c3
A__LENGTH :: tt:nil:cons:0':s:zeros:and:length:take -> c4:c5:c6
nil :: tt:nil:cons:0':s:zeros:and:length:take
c4 :: c4:c5:c6
cons :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c5 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c4:c5:c6
mark :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c6 :: c4:c5:c6
A__TAKE :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> c7:c8:c9
0' :: tt:nil:cons:0':s:zeros:and:length:take
c7 :: c7:c8:c9
s :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c8 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c7:c8:c9
c9 :: c7:c8:c9
zeros :: tt:nil:cons:0':s:zeros:and:length:take
c10 :: c:c1 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c11 :: c2:c3 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c12 :: c4:c5:c6 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
c13 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c14 :: c7:c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c15 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c16 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c17 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c18 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
c19 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19 -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
a__zeros :: tt:nil:cons:0':s:zeros:and:length:take
a__and :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__length :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
a__take :: tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take -> tt:nil:cons:0':s:zeros:and:length:take
hole_c:c11_20 :: c:c1
hole_c2:c32_20 :: c2:c3
hole_tt:nil:cons:0':s:zeros:and:length:take3_20 :: tt:nil:cons:0':s:zeros:and:length:take
hole_c10:c11:c12:c13:c14:c15:c16:c17:c18:c194_20 :: c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
hole_c4:c5:c65_20 :: c4:c5:c6
hole_c7:c8:c96_20 :: c7:c8:c9
gen_tt:nil:cons:0':s:zeros:and:length:take7_20 :: Nat -> tt:nil:cons:0':s:zeros:and:length:take
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20 :: Nat -> c10:c11:c12:c13:c14:c15:c16:c17:c18:c19
gen_c4:c5:c69_20 :: Nat -> c4:c5:c6


Lemmas:
a__length(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n11_20))) -> *10_20, rt in Omega(0)
A__LENGTH(gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(1, n39957_20))) -> *10_20, rt in Omega(n39957_20)


Generator Equations:
gen_tt:nil:cons:0':s:zeros:and:length:take7_20(0) <=> tt
gen_tt:nil:cons:0':s:zeros:and:length:take7_20(+(x, 1)) <=> cons(tt, gen_tt:nil:cons:0':s:zeros:and:length:take7_20(x))
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(0) <=> c10(c)
gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(+(x, 1)) <=> c11(c2(c10(c)), gen_c10:c11:c12:c13:c14:c15:c16:c17:c18:c198_20(x))
gen_c4:c5:c69_20(0) <=> c4
gen_c4:c5:c69_20(+(x, 1)) <=> c5(gen_c4:c5:c69_20(x), c10(c))


The following defined symbols remain to be analysed:
MARK

They will be analysed ascendingly in the following order:
MARK = A__LENGTH