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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 18 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 1193 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), 18.3 s]
        (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
   AND(tt, z0) -> c2(ACTIVATE(z0))
   LENGTH(nil) -> c3
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   TAKE(0, z0) -> c5
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   TAKE(z0, z1) -> c7
   ACTIVATE(n__zeros) -> c8(ZEROS)
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
   ACTIVATE(z0) -> c11

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

   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   and(tt, z0) -> activate(z0)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(activate(z1)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   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
   AND(tt, z0) -> c2(ACTIVATE(z0))
   LENGTH(nil) -> c3
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   TAKE(0', z0) -> c5
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   TAKE(z0, z1) -> c7
   ACTIVATE(n__zeros) -> c8(ZEROS)
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
   ACTIVATE(z0) -> c11

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

   zeros -> cons(0', n__zeros)
   zeros -> n__zeros
   and(tt, z0) -> activate(z0)
   length(nil) -> 0'
   length(cons(z0, z1)) -> s(length(activate(z1)))
   take(0', z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   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
AND(tt, z0) -> c2(ACTIVATE(z0))
LENGTH(nil) -> c3
LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
TAKE(0', z0) -> c5
TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
TAKE(z0, z1) -> c7
ACTIVATE(n__zeros) -> c8(ZEROS)
ACTIVATE(n__take(z0, z1)) -> c9(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) -> c10(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
ACTIVATE(z0) -> c11
zeros -> cons(0', n__zeros)
zeros -> n__zeros
and(tt, z0) -> activate(z0)
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(activate(z1)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
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
AND :: tt -> nil:cons:0':s:n__zeros:n__take -> c2
tt :: tt
c2 :: c8:c9:c10:c11 -> c2
ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c8:c9:c10:c11
LENGTH :: nil:cons:0':s:n__zeros:n__take -> c3:c4
nil :: nil:cons:0':s:n__zeros:n__take
c3 :: c3:c4
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
c4 :: c3:c4 -> c8:c9:c10:c11 -> c3:c4
activate :: 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 -> c5:c6:c7
0' :: nil:cons:0':s:n__zeros:n__take
c5 :: c5:c6:c7
s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c6 :: c8:c9:c10:c11 -> c5:c6:c7
c7 :: c5:c6:c7
n__zeros :: nil:cons:0':s:n__zeros:n__take
c8 :: c:c1 -> c8:c9:c10:c11
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
c9 :: c5:c6:c7 -> c8:c9:c10:c11 -> c8:c9:c10:c11
c10 :: c5:c6:c7 -> c8:c9:c10:c11 -> c8:c9:c10:c11
c11 :: c8:c9:c10:c11
zeros :: nil:cons:0':s:n__zeros:n__take
and :: 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
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_12 :: c:c1
hole_c22_12 :: c2
hole_tt3_12 :: tt
hole_nil:cons:0':s:n__zeros:n__take4_12 :: nil:cons:0':s:n__zeros:n__take
hole_c8:c9:c10:c115_12 :: c8:c9:c10:c11
hole_c3:c46_12 :: c3:c4
hole_c5:c6:c77_12 :: c5:c6:c7
gen_nil:cons:0':s:n__zeros:n__take8_12 :: Nat -> nil:cons:0':s:n__zeros:n__take
gen_c8:c9:c10:c119_12 :: Nat -> c8:c9:c10:c11
gen_c3:c410_12 :: Nat -> c3:c4

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

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

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

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

(6)
Obligation:
Innermost TRS:
Rules:
ZEROS -> c
ZEROS -> c1
AND(tt, z0) -> c2(ACTIVATE(z0))
LENGTH(nil) -> c3
LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
TAKE(0', z0) -> c5
TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
TAKE(z0, z1) -> c7
ACTIVATE(n__zeros) -> c8(ZEROS)
ACTIVATE(n__take(z0, z1)) -> c9(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) -> c10(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
ACTIVATE(z0) -> c11
zeros -> cons(0', n__zeros)
zeros -> n__zeros
and(tt, z0) -> activate(z0)
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(activate(z1)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
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
AND :: tt -> nil:cons:0':s:n__zeros:n__take -> c2
tt :: tt
c2 :: c8:c9:c10:c11 -> c2
ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c8:c9:c10:c11
LENGTH :: nil:cons:0':s:n__zeros:n__take -> c3:c4
nil :: nil:cons:0':s:n__zeros:n__take
c3 :: c3:c4
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
c4 :: c3:c4 -> c8:c9:c10:c11 -> c3:c4
activate :: 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 -> c5:c6:c7
0' :: nil:cons:0':s:n__zeros:n__take
c5 :: c5:c6:c7
s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c6 :: c8:c9:c10:c11 -> c5:c6:c7
c7 :: c5:c6:c7
n__zeros :: nil:cons:0':s:n__zeros:n__take
c8 :: c:c1 -> c8:c9:c10:c11
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
c9 :: c5:c6:c7 -> c8:c9:c10:c11 -> c8:c9:c10:c11
c10 :: c5:c6:c7 -> c8:c9:c10:c11 -> c8:c9:c10:c11
c11 :: c8:c9:c10:c11
zeros :: nil:cons:0':s:n__zeros:n__take
and :: 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
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_12 :: c:c1
hole_c22_12 :: c2
hole_tt3_12 :: tt
hole_nil:cons:0':s:n__zeros:n__take4_12 :: nil:cons:0':s:n__zeros:n__take
hole_c8:c9:c10:c115_12 :: c8:c9:c10:c11
hole_c3:c46_12 :: c3:c4
hole_c5:c6:c77_12 :: c5:c6:c7
gen_nil:cons:0':s:n__zeros:n__take8_12 :: Nat -> nil:cons:0':s:n__zeros:n__take
gen_c8:c9:c10:c119_12 :: Nat -> c8:c9:c10:c11
gen_c3:c410_12 :: Nat -> c3:c4


Generator Equations:
gen_nil:cons:0':s:n__zeros:n__take8_12(0) <=> nil
gen_nil:cons:0':s:n__zeros:n__take8_12(+(x, 1)) <=> cons(nil, gen_nil:cons:0':s:n__zeros:n__take8_12(x))
gen_c8:c9:c10:c119_12(0) <=> c8(c)
gen_c8:c9:c10:c119_12(+(x, 1)) <=> c9(c5, gen_c8:c9:c10:c119_12(x))
gen_c3:c410_12(0) <=> c3
gen_c3:c410_12(+(x, 1)) <=> c4(gen_c3:c410_12(x), c8(c))


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

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH(gen_nil:cons:0':s:n__zeros:n__take8_12(n49_12)) -> *11_12, rt in Omega(n49_12)

Induction Base:
LENGTH(gen_nil:cons:0':s:n__zeros:n__take8_12(0))

Induction Step:
LENGTH(gen_nil:cons:0':s:n__zeros:n__take8_12(+(n49_12, 1))) ->_R^Omega(1)
c4(LENGTH(activate(gen_nil:cons:0':s:n__zeros:n__take8_12(n49_12))), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take8_12(n49_12))) ->_R^Omega(0)
c4(LENGTH(gen_nil:cons:0':s:n__zeros:n__take8_12(n49_12)), ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take8_12(n49_12))) ->_IH
c4(*11_12, ACTIVATE(gen_nil:cons:0':s:n__zeros:n__take8_12(n49_12))) ->_R^Omega(1)
c4(*11_12, c11)

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
AND(tt, z0) -> c2(ACTIVATE(z0))
LENGTH(nil) -> c3
LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
TAKE(0', z0) -> c5
TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
TAKE(z0, z1) -> c7
ACTIVATE(n__zeros) -> c8(ZEROS)
ACTIVATE(n__take(z0, z1)) -> c9(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) -> c10(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
ACTIVATE(z0) -> c11
zeros -> cons(0', n__zeros)
zeros -> n__zeros
and(tt, z0) -> activate(z0)
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(activate(z1)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
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
AND :: tt -> nil:cons:0':s:n__zeros:n__take -> c2
tt :: tt
c2 :: c8:c9:c10:c11 -> c2
ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c8:c9:c10:c11
LENGTH :: nil:cons:0':s:n__zeros:n__take -> c3:c4
nil :: nil:cons:0':s:n__zeros:n__take
c3 :: c3:c4
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
c4 :: c3:c4 -> c8:c9:c10:c11 -> c3:c4
activate :: 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 -> c5:c6:c7
0' :: nil:cons:0':s:n__zeros:n__take
c5 :: c5:c6:c7
s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c6 :: c8:c9:c10:c11 -> c5:c6:c7
c7 :: c5:c6:c7
n__zeros :: nil:cons:0':s:n__zeros:n__take
c8 :: c:c1 -> c8:c9:c10:c11
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
c9 :: c5:c6:c7 -> c8:c9:c10:c11 -> c8:c9:c10:c11
c10 :: c5:c6:c7 -> c8:c9:c10:c11 -> c8:c9:c10:c11
c11 :: c8:c9:c10:c11
zeros :: nil:cons:0':s:n__zeros:n__take
and :: 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
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_12 :: c:c1
hole_c22_12 :: c2
hole_tt3_12 :: tt
hole_nil:cons:0':s:n__zeros:n__take4_12 :: nil:cons:0':s:n__zeros:n__take
hole_c8:c9:c10:c115_12 :: c8:c9:c10:c11
hole_c3:c46_12 :: c3:c4
hole_c5:c6:c77_12 :: c5:c6:c7
gen_nil:cons:0':s:n__zeros:n__take8_12 :: Nat -> nil:cons:0':s:n__zeros:n__take
gen_c8:c9:c10:c119_12 :: Nat -> c8:c9:c10:c11
gen_c3:c410_12 :: Nat -> c3:c4


Generator Equations:
gen_nil:cons:0':s:n__zeros:n__take8_12(0) <=> nil
gen_nil:cons:0':s:n__zeros:n__take8_12(+(x, 1)) <=> cons(nil, gen_nil:cons:0':s:n__zeros:n__take8_12(x))
gen_c8:c9:c10:c119_12(0) <=> c8(c)
gen_c8:c9:c10:c119_12(+(x, 1)) <=> c9(c5, gen_c8:c9:c10:c119_12(x))
gen_c3:c410_12(0) <=> c3
gen_c3:c410_12(+(x, 1)) <=> c4(gen_c3:c410_12(x), c8(c))


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
AND(tt, z0) -> c2(ACTIVATE(z0))
LENGTH(nil) -> c3
LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
TAKE(0', z0) -> c5
TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
TAKE(z0, z1) -> c7
ACTIVATE(n__zeros) -> c8(ZEROS)
ACTIVATE(n__take(z0, z1)) -> c9(TAKE(activate(z0), activate(z1)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) -> c10(TAKE(activate(z0), activate(z1)), ACTIVATE(z1))
ACTIVATE(z0) -> c11
zeros -> cons(0', n__zeros)
zeros -> n__zeros
and(tt, z0) -> activate(z0)
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(activate(z1)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
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
AND :: tt -> nil:cons:0':s:n__zeros:n__take -> c2
tt :: tt
c2 :: c8:c9:c10:c11 -> c2
ACTIVATE :: nil:cons:0':s:n__zeros:n__take -> c8:c9:c10:c11
LENGTH :: nil:cons:0':s:n__zeros:n__take -> c3:c4
nil :: nil:cons:0':s:n__zeros:n__take
c3 :: c3:c4
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
c4 :: c3:c4 -> c8:c9:c10:c11 -> c3:c4
activate :: 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 -> c5:c6:c7
0' :: nil:cons:0':s:n__zeros:n__take
c5 :: c5:c6:c7
s :: nil:cons:0':s:n__zeros:n__take -> nil:cons:0':s:n__zeros:n__take
c6 :: c8:c9:c10:c11 -> c5:c6:c7
c7 :: c5:c6:c7
n__zeros :: nil:cons:0':s:n__zeros:n__take
c8 :: c:c1 -> c8:c9:c10:c11
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
c9 :: c5:c6:c7 -> c8:c9:c10:c11 -> c8:c9:c10:c11
c10 :: c5:c6:c7 -> c8:c9:c10:c11 -> c8:c9:c10:c11
c11 :: c8:c9:c10:c11
zeros :: nil:cons:0':s:n__zeros:n__take
and :: 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
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_12 :: c:c1
hole_c22_12 :: c2
hole_tt3_12 :: tt
hole_nil:cons:0':s:n__zeros:n__take4_12 :: nil:cons:0':s:n__zeros:n__take
hole_c8:c9:c10:c115_12 :: c8:c9:c10:c11
hole_c3:c46_12 :: c3:c4
hole_c5:c6:c77_12 :: c5:c6:c7
gen_nil:cons:0':s:n__zeros:n__take8_12 :: Nat -> nil:cons:0':s:n__zeros:n__take
gen_c8:c9:c10:c119_12 :: Nat -> c8:c9:c10:c11
gen_c3:c410_12 :: Nat -> c3:c4


Lemmas:
LENGTH(gen_nil:cons:0':s:n__zeros:n__take8_12(n49_12)) -> *11_12, rt in Omega(n49_12)


Generator Equations:
gen_nil:cons:0':s:n__zeros:n__take8_12(0) <=> nil
gen_nil:cons:0':s:n__zeros:n__take8_12(+(x, 1)) <=> cons(nil, gen_nil:cons:0':s:n__zeros:n__take8_12(x))
gen_c8:c9:c10:c119_12(0) <=> c8(c)
gen_c8:c9:c10:c119_12(+(x, 1)) <=> c9(c5, gen_c8:c9:c10:c119_12(x))
gen_c3:c410_12(0) <=> c3
gen_c3:c410_12(+(x, 1)) <=> c4(gen_c3:c410_12(x), c8(c))


The following defined symbols remain to be analysed:
length
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons:0':s:n__zeros:n__take8_12(n5730_12)) -> *11_12, rt in Omega(0)

Induction Base:
length(gen_nil:cons:0':s:n__zeros:n__take8_12(0))

Induction Step:
length(gen_nil:cons:0':s:n__zeros:n__take8_12(+(n5730_12, 1))) ->_R^Omega(0)
s(length(activate(gen_nil:cons:0':s:n__zeros:n__take8_12(n5730_12)))) ->_R^Omega(0)
s(length(gen_nil:cons:0':s:n__zeros:n__take8_12(n5730_12))) ->_IH
s(*11_12)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(14)
BOUNDS(1, INF)