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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 16 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 573 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 78 ms]
        (14) BOUNDS(1, INF)


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

(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:

   ZEROS -> c
   ZEROS -> c1
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
   U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
   U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
   U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
   U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
   U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
   U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2))
   U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1))
   U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0))
   LENGTH(nil) -> c13
   LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1))
   TAKE(0, z0) -> c15
   TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2))
   TAKE(z0, z1) -> c17
   ACTIVATE(n__zeros) -> c18(ZEROS)
   ACTIVATE(n__take(z0, z1)) -> c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
   ACTIVATE(n__take(z0, z1)) -> c20(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
   ACTIVATE(z0) -> c21

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

   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   U11(tt, z0) -> U12(tt, activate(z0))
   U12(tt, z0) -> s(length(activate(z0)))
   U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2))
   U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2))
   U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0)))
   length(nil) -> 0
   length(cons(z0, z1)) -> U11(tt, activate(z1))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1)
   take(z0, z1) -> n__take(z0, z1)
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(activate(z0), activate(z1))
   activate(z0) -> 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:

   ZEROS -> c
   ZEROS -> c1
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
   U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
   U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
   U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
   U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
   U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
   U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2))
   U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1))
   U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0))
   LENGTH(nil) -> c13
   LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1))
   TAKE(0', z0) -> c15
   TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2))
   TAKE(z0, z1) -> c17
   ACTIVATE(n__zeros) -> c18(ZEROS)
   ACTIVATE(n__take(z0, z1)) -> c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
   ACTIVATE(n__take(z0, z1)) -> c20(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
   ACTIVATE(z0) -> c21

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

   zeros -> cons(0', n__zeros)
   zeros -> n__zeros
   U11(tt, z0) -> U12(tt, activate(z0))
   U12(tt, z0) -> s(length(activate(z0)))
   U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2))
   U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2))
   U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0)))
   length(nil) -> 0'
   length(cons(z0, z1)) -> U11(tt, activate(z1))
   take(0', z0) -> nil
   take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1)
   take(z0, z1) -> n__take(z0, z1)
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(activate(z0), activate(z1))
   activate(z0) -> z0

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

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

(4)
Obligation:
Innermost TRS:
Rules:
ZEROS -> c
ZEROS -> c1
U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2))
U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1))
U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0))
LENGTH(nil) -> c13
LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1))
TAKE(0', z0) -> c15
TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2))
TAKE(z0, z1) -> c17
ACTIVATE(n__zeros) -> c18(ZEROS)
ACTIVATE(n__take(z0, z1)) -> c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) -> c20(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
ACTIVATE(z0) -> c21
zeros -> cons(0', n__zeros)
zeros -> n__zeros
U11(tt, z0) -> U12(tt, activate(z0))
U12(tt, z0) -> s(length(activate(z0)))
U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2))
U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2))
U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> U11(tt, activate(z1))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1)
take(z0, z1) -> n__take(z0, z1)
activate(n__zeros) -> zeros
activate(n__take(z0, z1)) -> take(activate(z0), activate(z1))
activate(z0) -> z0

Types:
ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
U11' :: tt -> nil:cons:0':s:n__zeros:n__take -> c2
tt :: tt
c2 :: c3 -> c18:c19:c20:c21 -> c2
U12' :: tt -> nil:cons:0':s:n__zeros:n__take -> c3
activate :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c18:c19:c20:c21
c3 :: c13:c14 -> c18:c19:c20:c21 -> c3
LENGTH :: nil:cons:0':s:n__zeros:n__take -> c13:c14
U21' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c4:c5:c6
c4 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
U22' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c7:c8:c9
c5 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
c6 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
c7 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
U23' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c10:c11:c12
c8 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
c9 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
c10 :: c18:c19:c20:c21 -> c10:c11:c12
c11 :: c18:c19:c20:c21 -> c10:c11:c12
c12 :: c18:c19:c20:c21 -> c10:c11:c12
nil :: nil:cons:0':s:n__zeros:n__take
c13 :: c13:c14
cons :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c14 :: c2 -> c18:c19:c20:c21 -> c13:c14
TAKE :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c15:c16:c17
0' :: nil:cons:0':s:n__zeros:n__take
c15 :: c15:c16:c17
s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c16 :: c4:c5:c6 -> c18:c19:c20:c21 -> c15:c16:c17
c17 :: c15:c16:c17
n__zeros :: nil:cons:0':s:n__zeros:n__take
c18 :: c:c1 -> c18:c19:c20:c21
n__take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c19 :: c15:c16:c17 -> c18:c19:c20:c21 -> c18:c19:c20:c21
c20 :: c15:c16:c17 -> c18:c19:c20:c21 -> c18:c19:c20:c21
c21 :: c18:c19:c20:c21
zeros :: nil:cons:0':s:n__zeros:n__take
U11 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U12 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
length :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U21 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U22 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U23 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
hole_c:c11_24 :: c:c1
hole_c22_24 :: c2
hole_tt3_24 :: tt
hole_nil:cons:0':s:n__zeros:n__take4_24 :: nil:cons:0':s:n__zeros:n__take
hole_c35_24 :: c3
hole_c18:c19:c20:c216_24 :: c18:c19:c20:c21
hole_c13:c147_24 :: c13:c14
hole_c4:c5:c68_24 :: c4:c5:c6
hole_c7:c8:c99_24 :: c7:c8:c9
hole_c10:c11:c1210_24 :: c10:c11:c12
hole_c15:c16:c1711_24 :: c15:c16:c17
gen_nil:cons:0':s:n__zeros:n__take12_24 :: Nat -> nil:cons:0':s:n__zeros:n__take
gen_c18:c19:c20:c2113_24 :: Nat -> c18:c19:c20:c21

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
activate, ACTIVATE, LENGTH, length

They will be analysed ascendingly in the following order:
activate < ACTIVATE
activate < LENGTH
activate < length
ACTIVATE < LENGTH

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

(6)
Obligation:
Innermost TRS:
Rules:
ZEROS -> c
ZEROS -> c1
U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2))
U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1))
U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0))
LENGTH(nil) -> c13
LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1))
TAKE(0', z0) -> c15
TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2))
TAKE(z0, z1) -> c17
ACTIVATE(n__zeros) -> c18(ZEROS)
ACTIVATE(n__take(z0, z1)) -> c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) -> c20(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
ACTIVATE(z0) -> c21
zeros -> cons(0', n__zeros)
zeros -> n__zeros
U11(tt, z0) -> U12(tt, activate(z0))
U12(tt, z0) -> s(length(activate(z0)))
U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2))
U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2))
U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> U11(tt, activate(z1))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1)
take(z0, z1) -> n__take(z0, z1)
activate(n__zeros) -> zeros
activate(n__take(z0, z1)) -> take(activate(z0), activate(z1))
activate(z0) -> z0

Types:
ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
U11' :: tt -> nil:cons:0':s:n__zeros:n__take -> c2
tt :: tt
c2 :: c3 -> c18:c19:c20:c21 -> c2
U12' :: tt -> nil:cons:0':s:n__zeros:n__take -> c3
activate :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c18:c19:c20:c21
c3 :: c13:c14 -> c18:c19:c20:c21 -> c3
LENGTH :: nil:cons:0':s:n__zeros:n__take -> c13:c14
U21' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c4:c5:c6
c4 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
U22' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c7:c8:c9
c5 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
c6 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
c7 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
U23' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c10:c11:c12
c8 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
c9 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
c10 :: c18:c19:c20:c21 -> c10:c11:c12
c11 :: c18:c19:c20:c21 -> c10:c11:c12
c12 :: c18:c19:c20:c21 -> c10:c11:c12
nil :: nil:cons:0':s:n__zeros:n__take
c13 :: c13:c14
cons :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c14 :: c2 -> c18:c19:c20:c21 -> c13:c14
TAKE :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c15:c16:c17
0' :: nil:cons:0':s:n__zeros:n__take
c15 :: c15:c16:c17
s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c16 :: c4:c5:c6 -> c18:c19:c20:c21 -> c15:c16:c17
c17 :: c15:c16:c17
n__zeros :: nil:cons:0':s:n__zeros:n__take
c18 :: c:c1 -> c18:c19:c20:c21
n__take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c19 :: c15:c16:c17 -> c18:c19:c20:c21 -> c18:c19:c20:c21
c20 :: c15:c16:c17 -> c18:c19:c20:c21 -> c18:c19:c20:c21
c21 :: c18:c19:c20:c21
zeros :: nil:cons:0':s:n__zeros:n__take
U11 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U12 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
length :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U21 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U22 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U23 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
hole_c:c11_24 :: c:c1
hole_c22_24 :: c2
hole_tt3_24 :: tt
hole_nil:cons:0':s:n__zeros:n__take4_24 :: nil:cons:0':s:n__zeros:n__take
hole_c35_24 :: c3
hole_c18:c19:c20:c216_24 :: c18:c19:c20:c21
hole_c13:c147_24 :: c13:c14
hole_c4:c5:c68_24 :: c4:c5:c6
hole_c7:c8:c99_24 :: c7:c8:c9
hole_c10:c11:c1210_24 :: c10:c11:c12
hole_c15:c16:c1711_24 :: c15:c16:c17
gen_nil:cons:0':s:n__zeros:n__take12_24 :: Nat -> nil:cons:0':s:n__zeros:n__take
gen_c18:c19:c20:c2113_24 :: Nat -> c18:c19:c20:c21


Generator Equations:
gen_nil:cons:0':s:n__zeros:n__take12_24(0) <=> nil
gen_nil:cons:0':s:n__zeros:n__take12_24(+(x, 1)) <=> cons(nil, gen_nil:cons:0':s:n__zeros:n__take12_24(x))
gen_c18:c19:c20:c2113_24(0) <=> c18(c)
gen_c18:c19:c20:c2113_24(+(x, 1)) <=> c19(c15, gen_c18:c19:c20:c2113_24(x))


The following defined symbols remain to be analysed:
activate, ACTIVATE, LENGTH, length

They will be analysed ascendingly in the following order:
activate < ACTIVATE
activate < LENGTH
activate < length
ACTIVATE < LENGTH

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24)) -> *14_24, rt in Omega(n52_24)

Induction Base:
LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(0))

Induction Step:
LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(+(n52_24, 1))) ->_R^Omega(1)
c14(U11'(tt, activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))) ->_R^Omega(0)
c14(U11'(tt, gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24)), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))) ->_R^Omega(1)
c14(c2(U12'(tt, activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))) ->_R^Omega(0)
c14(c2(U12'(tt, gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24)), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))) ->_R^Omega(1)
c14(c2(c3(LENGTH(activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))) ->_R^Omega(0)
c14(c2(c3(LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24)), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))) ->_IH
c14(c2(c3(*14_24, ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))) ->_R^Omega(1)
c14(c2(c3(*14_24, c21), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))) ->_R^Omega(1)
c14(c2(c3(*14_24, c21), c21), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24))) ->_R^Omega(1)
c14(c2(c3(*14_24, c21), c21), c21)

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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
ZEROS -> c
ZEROS -> c1
U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2))
U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1))
U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0))
LENGTH(nil) -> c13
LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1))
TAKE(0', z0) -> c15
TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2))
TAKE(z0, z1) -> c17
ACTIVATE(n__zeros) -> c18(ZEROS)
ACTIVATE(n__take(z0, z1)) -> c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) -> c20(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
ACTIVATE(z0) -> c21
zeros -> cons(0', n__zeros)
zeros -> n__zeros
U11(tt, z0) -> U12(tt, activate(z0))
U12(tt, z0) -> s(length(activate(z0)))
U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2))
U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2))
U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> U11(tt, activate(z1))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1)
take(z0, z1) -> n__take(z0, z1)
activate(n__zeros) -> zeros
activate(n__take(z0, z1)) -> take(activate(z0), activate(z1))
activate(z0) -> z0

Types:
ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
U11' :: tt -> nil:cons:0':s:n__zeros:n__take -> c2
tt :: tt
c2 :: c3 -> c18:c19:c20:c21 -> c2
U12' :: tt -> nil:cons:0':s:n__zeros:n__take -> c3
activate :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c18:c19:c20:c21
c3 :: c13:c14 -> c18:c19:c20:c21 -> c3
LENGTH :: nil:cons:0':s:n__zeros:n__take -> c13:c14
U21' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c4:c5:c6
c4 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
U22' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c7:c8:c9
c5 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
c6 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
c7 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
U23' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c10:c11:c12
c8 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
c9 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
c10 :: c18:c19:c20:c21 -> c10:c11:c12
c11 :: c18:c19:c20:c21 -> c10:c11:c12
c12 :: c18:c19:c20:c21 -> c10:c11:c12
nil :: nil:cons:0':s:n__zeros:n__take
c13 :: c13:c14
cons :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c14 :: c2 -> c18:c19:c20:c21 -> c13:c14
TAKE :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c15:c16:c17
0' :: nil:cons:0':s:n__zeros:n__take
c15 :: c15:c16:c17
s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c16 :: c4:c5:c6 -> c18:c19:c20:c21 -> c15:c16:c17
c17 :: c15:c16:c17
n__zeros :: nil:cons:0':s:n__zeros:n__take
c18 :: c:c1 -> c18:c19:c20:c21
n__take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c19 :: c15:c16:c17 -> c18:c19:c20:c21 -> c18:c19:c20:c21
c20 :: c15:c16:c17 -> c18:c19:c20:c21 -> c18:c19:c20:c21
c21 :: c18:c19:c20:c21
zeros :: nil:cons:0':s:n__zeros:n__take
U11 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U12 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
length :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U21 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U22 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U23 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
hole_c:c11_24 :: c:c1
hole_c22_24 :: c2
hole_tt3_24 :: tt
hole_nil:cons:0':s:n__zeros:n__take4_24 :: nil:cons:0':s:n__zeros:n__take
hole_c35_24 :: c3
hole_c18:c19:c20:c216_24 :: c18:c19:c20:c21
hole_c13:c147_24 :: c13:c14
hole_c4:c5:c68_24 :: c4:c5:c6
hole_c7:c8:c99_24 :: c7:c8:c9
hole_c10:c11:c1210_24 :: c10:c11:c12
hole_c15:c16:c1711_24 :: c15:c16:c17
gen_nil:cons:0':s:n__zeros:n__take12_24 :: Nat -> nil:cons:0':s:n__zeros:n__take
gen_c18:c19:c20:c2113_24 :: Nat -> c18:c19:c20:c21


Generator Equations:
gen_nil:cons:0':s:n__zeros:n__take12_24(0) <=> nil
gen_nil:cons:0':s:n__zeros:n__take12_24(+(x, 1)) <=> cons(nil, gen_nil:cons:0':s:n__zeros:n__take12_24(x))
gen_c18:c19:c20:c2113_24(0) <=> c18(c)
gen_c18:c19:c20:c2113_24(+(x, 1)) <=> c19(c15, gen_c18:c19:c20:c2113_24(x))


The following defined symbols remain to be analysed:
LENGTH, length
----------------------------------------

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
ZEROS -> c
ZEROS -> c1
U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1, z2) -> c4(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
U21'(tt, z0, z1, z2) -> c5(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
U21'(tt, z0, z1, z2) -> c6(U22'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
U22'(tt, z0, z1, z2) -> c7(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z0))
U22'(tt, z0, z1, z2) -> c8(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z1))
U22'(tt, z0, z1, z2) -> c9(U23'(tt, activate(z0), activate(z1), activate(z2)), ACTIVATE(z2))
U23'(tt, z0, z1, z2) -> c10(ACTIVATE(z2))
U23'(tt, z0, z1, z2) -> c11(ACTIVATE(z1))
U23'(tt, z0, z1, z2) -> c12(ACTIVATE(z0))
LENGTH(nil) -> c13
LENGTH(cons(z0, z1)) -> c14(U11'(tt, activate(z1)), ACTIVATE(z1))
TAKE(0', z0) -> c15
TAKE(s(z0), cons(z1, z2)) -> c16(U21'(tt, activate(z2), z0, z1), ACTIVATE(z2))
TAKE(z0, z1) -> c17
ACTIVATE(n__zeros) -> c18(ZEROS)
ACTIVATE(n__take(z0, z1)) -> c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) -> c20(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
ACTIVATE(z0) -> c21
zeros -> cons(0', n__zeros)
zeros -> n__zeros
U11(tt, z0) -> U12(tt, activate(z0))
U12(tt, z0) -> s(length(activate(z0)))
U21(tt, z0, z1, z2) -> U22(tt, activate(z0), activate(z1), activate(z2))
U22(tt, z0, z1, z2) -> U23(tt, activate(z0), activate(z1), activate(z2))
U23(tt, z0, z1, z2) -> cons(activate(z2), n__take(activate(z1), activate(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> U11(tt, activate(z1))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> U21(tt, activate(z2), z0, z1)
take(z0, z1) -> n__take(z0, z1)
activate(n__zeros) -> zeros
activate(n__take(z0, z1)) -> take(activate(z0), activate(z1))
activate(z0) -> z0

Types:
ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
U11' :: tt -> nil:cons:0':s:n__zeros:n__take -> c2
tt :: tt
c2 :: c3 -> c18:c19:c20:c21 -> c2
U12' :: tt -> nil:cons:0':s:n__zeros:n__take -> c3
activate :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c18:c19:c20:c21
c3 :: c13:c14 -> c18:c19:c20:c21 -> c3
LENGTH :: nil:cons:0':s:n__zeros:n__take -> c13:c14
U21' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c4:c5:c6
c4 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
U22' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c7:c8:c9
c5 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
c6 :: c7:c8:c9 -> c18:c19:c20:c21 -> c4:c5:c6
c7 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
U23' :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c10:c11:c12
c8 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
c9 :: c10:c11:c12 -> c18:c19:c20:c21 -> c7:c8:c9
c10 :: c18:c19:c20:c21 -> c10:c11:c12
c11 :: c18:c19:c20:c21 -> c10:c11:c12
c12 :: c18:c19:c20:c21 -> c10:c11:c12
nil :: nil:cons:0':s:n__zeros:n__take
c13 :: c13:c14
cons :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c14 :: c2 -> c18:c19:c20:c21 -> c13:c14
TAKE :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> c15:c16:c17
0' :: nil:cons:0':s:n__zeros:n__take
c15 :: c15:c16:c17
s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c16 :: c4:c5:c6 -> c18:c19:c20:c21 -> c15:c16:c17
c17 :: c15:c16:c17
n__zeros :: nil:cons:0':s:n__zeros:n__take
c18 :: c:c1 -> c18:c19:c20:c21
n__take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c19 :: c15:c16:c17 -> c18:c19:c20:c21 -> c18:c19:c20:c21
c20 :: c15:c16:c17 -> c18:c19:c20:c21 -> c18:c19:c20:c21
c21 :: c18:c19:c20:c21
zeros :: nil:cons:0':s:n__zeros:n__take
U11 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U12 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
length :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U21 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U22 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
U23 :: tt -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
take :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
hole_c:c11_24 :: c:c1
hole_c22_24 :: c2
hole_tt3_24 :: tt
hole_nil:cons:0':s:n__zeros:n__take4_24 :: nil:cons:0':s:n__zeros:n__take
hole_c35_24 :: c3
hole_c18:c19:c20:c216_24 :: c18:c19:c20:c21
hole_c13:c147_24 :: c13:c14
hole_c4:c5:c68_24 :: c4:c5:c6
hole_c7:c8:c99_24 :: c7:c8:c9
hole_c10:c11:c1210_24 :: c10:c11:c12
hole_c15:c16:c1711_24 :: c15:c16:c17
gen_nil:cons:0':s:n__zeros:n__take12_24 :: Nat -> nil:cons:0':s:n__zeros:n__take
gen_c18:c19:c20:c2113_24 :: Nat -> c18:c19:c20:c21


Lemmas:
LENGTH(gen_nil:cons:0':s:n__zeros:n__take12_24(n52_24)) -> *14_24, rt in Omega(n52_24)


Generator Equations:
gen_nil:cons:0':s:n__zeros:n__take12_24(0) <=> nil
gen_nil:cons:0':s:n__zeros:n__take12_24(+(x, 1)) <=> cons(nil, gen_nil:cons:0':s:n__zeros:n__take12_24(x))
gen_c18:c19:c20:c2113_24(0) <=> c18(c)
gen_c18:c19:c20:c2113_24(+(x, 1)) <=> c19(c15, gen_c18:c19:c20:c2113_24(x))


The following defined symbols remain to be analysed:
length
----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons:0':s:n__zeros:n__take12_24(n2514_24)) -> *14_24, rt in Omega(0)

Induction Base:
length(gen_nil:cons:0':s:n__zeros:n__take12_24(0))

Induction Step:
length(gen_nil:cons:0':s:n__zeros:n__take12_24(+(n2514_24, 1))) ->_R^Omega(0)
U11(tt, activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n2514_24))) ->_R^Omega(0)
U11(tt, gen_nil:cons:0':s:n__zeros:n__take12_24(n2514_24)) ->_R^Omega(0)
U12(tt, activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n2514_24))) ->_R^Omega(0)
U12(tt, gen_nil:cons:0':s:n__zeros:n__take12_24(n2514_24)) ->_R^Omega(0)
s(length(activate(gen_nil:cons:0':s:n__zeros:n__take12_24(n2514_24)))) ->_R^Omega(0)
s(length(gen_nil:cons:0':s:n__zeros:n__take12_24(n2514_24))) ->_IH
s(*14_24)

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

(14)
BOUNDS(1, INF)