### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
sieve(mark(X)) →+ mark(sieve(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
active, sieve, from, s, cons, if, divides, filter, head, tail, proper, top

They will be analysed ascendingly in the following order:
sieve < active
from < active
s < active
cons < active
if < active
divides < active
filter < active
tail < active
active < top
sieve < proper
from < proper
s < proper
cons < proper
if < proper
divides < proper
filter < proper
tail < proper
proper < top

### (8) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
sieve, active, from, s, cons, if, divides, filter, head, tail, proper, top

They will be analysed ascendingly in the following order:
sieve < active
from < active
s < active
cons < active
if < active
divides < active
filter < active
tail < active
active < top
sieve < proper
from < proper
s < proper
cons < proper
if < proper
divides < proper
filter < proper
tail < proper
proper < top

### (9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

Induction Base:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, +(n5_0, 1)))) →RΩ(1)
mark(sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (11) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
from, active, s, cons, if, divides, filter, head, tail, proper, top

They will be analysed ascendingly in the following order:
from < active
s < active
cons < active
if < active
divides < active
filter < active
tail < active
active < top
from < proper
s < proper
cons < proper
if < proper
divides < proper
filter < proper
tail < proper
proper < top

### (12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)

Induction Base:
from(gen_primes:0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
from(gen_primes:0':mark:true:false:ok3_0(+(1, +(n434_0, 1)))) →RΩ(1)
mark(from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (14) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
s, active, cons, if, divides, filter, head, tail, proper, top

They will be analysed ascendingly in the following order:
s < active
cons < active
if < active
divides < active
filter < active
tail < active
active < top
s < proper
cons < proper
if < proper
divides < proper
filter < proper
tail < proper
proper < top

### (15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)

Induction Base:
s(gen_primes:0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
s(gen_primes:0':mark:true:false:ok3_0(+(1, +(n964_0, 1)))) →RΩ(1)
mark(s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (17) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, if, divides, filter, head, tail, proper, top

They will be analysed ascendingly in the following order:
cons < active
if < active
divides < active
filter < active
tail < active
active < top
cons < proper
if < proper
divides < proper
filter < proper
tail < proper
proper < top

### (18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)

Induction Base:
cons(gen_primes:0':mark:true:false:ok3_0(+(1, 0)), gen_primes:0':mark:true:false:ok3_0(b))

Induction Step:
cons(gen_primes:0':mark:true:false:ok3_0(+(1, +(n1595_0, 1))), gen_primes:0':mark:true:false:ok3_0(b)) →RΩ(1)
mark(cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (20) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
if, active, divides, filter, head, tail, proper, top

They will be analysed ascendingly in the following order:
if < active
divides < active
filter < active
tail < active
active < top
if < proper
divides < proper
filter < proper
tail < proper
proper < top

### (21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)

Induction Base:
if(gen_primes:0':mark:true:false:ok3_0(+(1, 0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c))

Induction Step:
if(gen_primes:0':mark:true:false:ok3_0(+(1, +(n3398_0, 1))), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) →RΩ(1)
mark(if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (23) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
divides, active, filter, head, tail, proper, top

They will be analysed ascendingly in the following order:
divides < active
filter < active
tail < active
active < top
divides < proper
filter < proper
tail < proper
proper < top

### (24) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)

Induction Base:
divides(gen_primes:0':mark:true:false:ok3_0(+(1, 0)), gen_primes:0':mark:true:false:ok3_0(b))

Induction Step:
divides(gen_primes:0':mark:true:false:ok3_0(+(1, +(n6834_0, 1))), gen_primes:0':mark:true:false:ok3_0(b)) →RΩ(1)
mark(divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (26) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
filter, active, head, tail, proper, top

They will be analysed ascendingly in the following order:
filter < active
tail < active
active < top
filter < proper
tail < proper
proper < top

### (27) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)

Induction Base:
filter(gen_primes:0':mark:true:false:ok3_0(+(1, 0)), gen_primes:0':mark:true:false:ok3_0(b))

Induction Step:
filter(gen_primes:0':mark:true:false:ok3_0(+(1, +(n9542_0, 1))), gen_primes:0':mark:true:false:ok3_0(b)) →RΩ(1)
mark(filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (29) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
tail < active
active < top
tail < proper
proper < top

### (30) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
head(gen_primes:0':mark:true:false:ok3_0(+(1, n12554_0))) → *4_0, rt ∈ Ω(n125540)

Induction Base:

Induction Step:
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (32) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)
head(gen_primes:0':mark:true:false:ok3_0(+(1, n12554_0))) → *4_0, rt ∈ Ω(n125540)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
tail, active, proper, top

They will be analysed ascendingly in the following order:
tail < active
active < top
tail < proper
proper < top

### (33) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
tail(gen_primes:0':mark:true:false:ok3_0(+(1, n13935_0))) → *4_0, rt ∈ Ω(n139350)

Induction Base:
tail(gen_primes:0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
tail(gen_primes:0':mark:true:false:ok3_0(+(1, +(n13935_0, 1)))) →RΩ(1)
mark(tail(gen_primes:0':mark:true:false:ok3_0(+(1, n13935_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (35) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)
head(gen_primes:0':mark:true:false:ok3_0(+(1, n12554_0))) → *4_0, rt ∈ Ω(n125540)
tail(gen_primes:0':mark:true:false:ok3_0(+(1, n13935_0))) → *4_0, rt ∈ Ω(n139350)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top

### (36) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol active.

### (37) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)
head(gen_primes:0':mark:true:false:ok3_0(+(1, n12554_0))) → *4_0, rt ∈ Ω(n125540)
tail(gen_primes:0':mark:true:false:ok3_0(+(1, n13935_0))) → *4_0, rt ∈ Ω(n139350)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
proper, top

They will be analysed ascendingly in the following order:
proper < top

### (38) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol proper.

### (39) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)
head(gen_primes:0':mark:true:false:ok3_0(+(1, n12554_0))) → *4_0, rt ∈ Ω(n125540)
tail(gen_primes:0':mark:true:false:ok3_0(+(1, n13935_0))) → *4_0, rt ∈ Ω(n139350)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
top

### (40) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol top.

### (41) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)
head(gen_primes:0':mark:true:false:ok3_0(+(1, n12554_0))) → *4_0, rt ∈ Ω(n125540)
tail(gen_primes:0':mark:true:false:ok3_0(+(1, n13935_0))) → *4_0, rt ∈ Ω(n139350)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (42) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (44) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)
head(gen_primes:0':mark:true:false:ok3_0(+(1, n12554_0))) → *4_0, rt ∈ Ω(n125540)
tail(gen_primes:0':mark:true:false:ok3_0(+(1, n13935_0))) → *4_0, rt ∈ Ω(n139350)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (45) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (47) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)
head(gen_primes:0':mark:true:false:ok3_0(+(1, n12554_0))) → *4_0, rt ∈ Ω(n125540)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (48) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (50) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)
filter(gen_primes:0':mark:true:false:ok3_0(+(1, n9542_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n95420)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (51) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (53) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)
divides(gen_primes:0':mark:true:false:ok3_0(+(1, n6834_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n68340)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (54) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (56) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)
if(gen_primes:0':mark:true:false:ok3_0(+(1, n3398_0)), gen_primes:0':mark:true:false:ok3_0(b), gen_primes:0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n33980)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (57) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (59) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)
cons(gen_primes:0':mark:true:false:ok3_0(+(1, n1595_0)), gen_primes:0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n15950)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (60) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (62) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)
s(gen_primes:0':mark:true:false:ok3_0(+(1, n964_0))) → *4_0, rt ∈ Ω(n9640)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (63) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (65) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
from(gen_primes:0':mark:true:false:ok3_0(+(1, n434_0))) → *4_0, rt ∈ Ω(n4340)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (66) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (68) Obligation:

TRS:
Rules:
active(primes) → mark(sieve(from(s(s(0')))))
active(from(X)) → mark(cons(X, from(s(X))))
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
primes :: primes:0':mark:true:false:ok
mark :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
sieve :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
from :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
s :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
0' :: primes:0':mark:true:false:ok
cons :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
tail :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
if :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
true :: primes:0':mark:true:false:ok
false :: primes:0':mark:true:false:ok
filter :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
divides :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
proper :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
ok :: primes:0':mark:true:false:ok → primes:0':mark:true:false:ok
top :: primes:0':mark:true:false:ok → top
hole_primes:0':mark:true:false:ok1_0 :: primes:0':mark:true:false:ok
hole_top2_0 :: top
gen_primes:0':mark:true:false:ok3_0 :: Nat → primes:0':mark:true:false:ok

Lemmas:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_primes:0':mark:true:false:ok3_0(0) ⇔ primes
gen_primes:0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_primes:0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

### (69) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sieve(gen_primes:0':mark:true:false:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)