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

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2ndspos'(0', Z)) → mark'(rnil')
active'(2ndspos'(s'(N), cons'(X, cons'(Y, Z)))) → mark'(rcons'(posrecip'(Y), 2ndsneg'(N, Z)))
active'(2ndsneg'(0', Z)) → mark'(rnil')
active'(2ndsneg'(s'(N), cons'(X, cons'(Y, Z)))) → mark'(rcons'(negrecip'(Y), 2ndspos'(N, Z)))
active'(pi'(X)) → mark'(2ndspos'(X, from'(0')))
active'(plus'(0', Y)) → mark'(Y)
active'(plus'(s'(X), Y)) → mark'(s'(plus'(X, Y)))
active'(times'(0', Y)) → mark'(0')
active'(times'(s'(X), Y)) → mark'(plus'(Y, times'(X, Y)))
active'(square'(X)) → mark'(times'(X, X))
active'(s'(X)) → s'(active'(X))
active'(posrecip'(X)) → posrecip'(active'(X))
active'(negrecip'(X)) → negrecip'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(rcons'(X1, X2)) → rcons'(active'(X1), X2)
active'(rcons'(X1, X2)) → rcons'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(2ndspos'(X1, X2)) → 2ndspos'(active'(X1), X2)
active'(2ndspos'(X1, X2)) → 2ndspos'(X1, active'(X2))
active'(2ndsneg'(X1, X2)) → 2ndsneg'(active'(X1), X2)
active'(2ndsneg'(X1, X2)) → 2ndsneg'(X1, active'(X2))
active'(pi'(X)) → pi'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
active'(times'(X1, X2)) → times'(active'(X1), X2)
active'(times'(X1, X2)) → times'(X1, active'(X2))
active'(square'(X)) → square'(active'(X))
s'(mark'(X)) → mark'(s'(X))
posrecip'(mark'(X)) → mark'(posrecip'(X))
negrecip'(mark'(X)) → mark'(negrecip'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
rcons'(mark'(X1), X2) → mark'(rcons'(X1, X2))
rcons'(X1, mark'(X2)) → mark'(rcons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
2ndspos'(mark'(X1), X2) → mark'(2ndspos'(X1, X2))
2ndspos'(X1, mark'(X2)) → mark'(2ndspos'(X1, X2))
2ndsneg'(mark'(X1), X2) → mark'(2ndsneg'(X1, X2))
2ndsneg'(X1, mark'(X2)) → mark'(2ndsneg'(X1, X2))
pi'(mark'(X)) → mark'(pi'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
times'(mark'(X1), X2) → mark'(times'(X1, X2))
times'(X1, mark'(X2)) → mark'(times'(X1, X2))
square'(mark'(X)) → mark'(square'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(posrecip'(X)) → posrecip'(proper'(X))
proper'(negrecip'(X)) → negrecip'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(rnil') → ok'(rnil')
proper'(rcons'(X1, X2)) → rcons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(2ndspos'(X1, X2)) → 2ndspos'(proper'(X1), proper'(X2))
proper'(2ndsneg'(X1, X2)) → 2ndsneg'(proper'(X1), proper'(X2))
proper'(pi'(X)) → pi'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(times'(X1, X2)) → times'(proper'(X1), proper'(X2))
proper'(square'(X)) → square'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
posrecip'(ok'(X)) → ok'(posrecip'(X))
negrecip'(ok'(X)) → ok'(negrecip'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
rcons'(ok'(X1), ok'(X2)) → ok'(rcons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
2ndspos'(ok'(X1), ok'(X2)) → ok'(2ndspos'(X1, X2))
2ndsneg'(ok'(X1), ok'(X2)) → ok'(2ndsneg'(X1, X2))
pi'(ok'(X)) → ok'(pi'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
times'(ok'(X1), ok'(X2)) → ok'(times'(X1, X2))
square'(ok'(X)) → ok'(square'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2ndspos'(0', Z)) → mark'(rnil')
active'(2ndspos'(s'(N), cons'(X, cons'(Y, Z)))) → mark'(rcons'(posrecip'(Y), 2ndsneg'(N, Z)))
active'(2ndsneg'(0', Z)) → mark'(rnil')
active'(2ndsneg'(s'(N), cons'(X, cons'(Y, Z)))) → mark'(rcons'(negrecip'(Y), 2ndspos'(N, Z)))
active'(pi'(X)) → mark'(2ndspos'(X, from'(0')))
active'(plus'(0', Y)) → mark'(Y)
active'(plus'(s'(X), Y)) → mark'(s'(plus'(X, Y)))
active'(times'(0', Y)) → mark'(0')
active'(times'(s'(X), Y)) → mark'(plus'(Y, times'(X, Y)))
active'(square'(X)) → mark'(times'(X, X))
active'(s'(X)) → s'(active'(X))
active'(posrecip'(X)) → posrecip'(active'(X))
active'(negrecip'(X)) → negrecip'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(rcons'(X1, X2)) → rcons'(active'(X1), X2)
active'(rcons'(X1, X2)) → rcons'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(2ndspos'(X1, X2)) → 2ndspos'(active'(X1), X2)
active'(2ndspos'(X1, X2)) → 2ndspos'(X1, active'(X2))
active'(2ndsneg'(X1, X2)) → 2ndsneg'(active'(X1), X2)
active'(2ndsneg'(X1, X2)) → 2ndsneg'(X1, active'(X2))
active'(pi'(X)) → pi'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
active'(times'(X1, X2)) → times'(active'(X1), X2)
active'(times'(X1, X2)) → times'(X1, active'(X2))
active'(square'(X)) → square'(active'(X))
s'(mark'(X)) → mark'(s'(X))
posrecip'(mark'(X)) → mark'(posrecip'(X))
negrecip'(mark'(X)) → mark'(negrecip'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
rcons'(mark'(X1), X2) → mark'(rcons'(X1, X2))
rcons'(X1, mark'(X2)) → mark'(rcons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
2ndspos'(mark'(X1), X2) → mark'(2ndspos'(X1, X2))
2ndspos'(X1, mark'(X2)) → mark'(2ndspos'(X1, X2))
2ndsneg'(mark'(X1), X2) → mark'(2ndsneg'(X1, X2))
2ndsneg'(X1, mark'(X2)) → mark'(2ndsneg'(X1, X2))
pi'(mark'(X)) → mark'(pi'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
times'(mark'(X1), X2) → mark'(times'(X1, X2))
times'(X1, mark'(X2)) → mark'(times'(X1, X2))
square'(mark'(X)) → mark'(square'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(posrecip'(X)) → posrecip'(proper'(X))
proper'(negrecip'(X)) → negrecip'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(rnil') → ok'(rnil')
proper'(rcons'(X1, X2)) → rcons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(2ndspos'(X1, X2)) → 2ndspos'(proper'(X1), proper'(X2))
proper'(2ndsneg'(X1, X2)) → 2ndsneg'(proper'(X1), proper'(X2))
proper'(pi'(X)) → pi'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(times'(X1, X2)) → times'(proper'(X1), proper'(X2))
proper'(square'(X)) → square'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
posrecip'(ok'(X)) → ok'(posrecip'(X))
negrecip'(ok'(X)) → ok'(negrecip'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
rcons'(ok'(X1), ok'(X2)) → ok'(rcons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
2ndspos'(ok'(X1), ok'(X2)) → ok'(2ndspos'(X1, X2))
2ndsneg'(ok'(X1), ok'(X2)) → ok'(2ndsneg'(X1, X2))
pi'(ok'(X)) → ok'(pi'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
times'(ok'(X1), ok'(X2)) → ok'(times'(X1, X2))
square'(ok'(X)) → ok'(square'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
from' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
mark' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
cons' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
s' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
2ndspos' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
0' :: mark':0':rnil':ok':nil'
rnil' :: mark':0':rnil':ok':nil'
rcons' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
posrecip' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
2ndsneg' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
negrecip' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
pi' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
plus' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
times' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
square' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
proper' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
ok' :: mark':0':rnil':ok':nil' → mark':0':rnil':ok':nil'
nil' :: mark':0':rnil':ok':nil'
top' :: mark':0':rnil':ok':nil' → top'
_hole_mark':0':rnil':ok':nil'1 :: mark':0':rnil':ok':nil'
_hole_top'2 :: top'
_gen_mark':0':rnil':ok':nil'3 :: Nat → mark':0':rnil':ok':nil'

Heuristically decided to analyse the following defined symbols:
active', cons', from', s', rcons', posrecip', 2ndsneg', negrecip', 2ndspos', plus', times', pi', square', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
from' < active'
s' < active'
rcons' < active'
posrecip' < active'
2ndsneg' < active'
negrecip' < active'
2ndspos' < active'
plus' < active'
times' < active'
pi' < active'
square' < active'
active' < top'
cons' < proper'
from' < proper'
s' < proper'
rcons' < proper'
posrecip' < proper'
2ndsneg' < proper'
negrecip' < proper'
2ndspos' < proper'
plus' < proper'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
cons', active', from', s', rcons', posrecip', 2ndsneg', negrecip', 2ndspos', plus', times', pi', square', proper', top'

Proved the following rewrite lemma:
cons'(_gen_mark':0':rnil':ok':nil'3(+(1, _n5)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n5)

Induction Base:
cons'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)), _gen_mark':0':rnil':ok':nil'3(b))

Induction Step:
cons'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n6, 1))), _gen_mark':0':rnil':ok':nil'3(_b610)) →_R^Ω(1)
mark'(cons'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n6)), _gen_mark':0':rnil':ok':nil'3(_b610))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_mark':0':rnil':ok':nil'3(+(1, _n5)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
from', active', s', rcons', posrecip', 2ndsneg', negrecip', 2ndspos', plus', times', pi', square', proper', top'

They will be analysed ascendingly in the following order:
from' < active'
s' < active'
rcons' < active'
posrecip' < active'
2ndsneg' < active'
negrecip' < active'
2ndspos' < active'
plus' < active'
times' < active'
pi' < active'
square' < active'
active' < top'
from' < proper'
s' < proper'
rcons' < proper'
posrecip' < proper'
2ndsneg' < proper'
negrecip' < proper'
2ndspos' < proper'
plus' < proper'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Proved the following rewrite lemma:
from'(_gen_mark':0':rnil':ok':nil'3(+(1, _n3878))) → _*4, rt ∈ Ω(_n3878)

Induction Base:
from'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)))

Induction Step:
from'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n3879, 1)))) →_R^Ω(1)
mark'(from'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n3879)))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_mark':0':rnil':ok':nil'3(+(1, _n5)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n5)
from'(_gen_mark':0':rnil':ok':nil'3(+(1, _n3878))) → _*4, rt ∈ Ω(_n3878)

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
s', active', rcons', posrecip', 2ndsneg', negrecip', 2ndspos', plus', times', pi', square', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
rcons' < active'
posrecip' < active'
2ndsneg' < active'
negrecip' < active'
2ndspos' < active'
plus' < active'
times' < active'
pi' < active'
square' < active'
active' < top'
s' < proper'
rcons' < proper'
posrecip' < proper'
2ndsneg' < proper'
negrecip' < proper'
2ndspos' < proper'
plus' < proper'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_mark':0':rnil':ok':nil'3(+(1, _n6516))) → _*4, rt ∈ Ω(_n6516)

Induction Base:
s'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)))

Induction Step:
s'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n6517, 1)))) →_R^Ω(1)
mark'(s'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n6517)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
rcons', active', posrecip', 2ndsneg', negrecip', 2ndspos', plus', times', pi', square', proper', top'

They will be analysed ascendingly in the following order:
rcons' < active'
posrecip' < active'
2ndsneg' < active'
negrecip' < active'
2ndspos' < active'
plus' < active'
times' < active'
pi' < active'
square' < active'
active' < top'
rcons' < proper'
posrecip' < proper'
2ndsneg' < proper'
negrecip' < proper'
2ndspos' < proper'
plus' < proper'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Proved the following rewrite lemma:
rcons'(_gen_mark':0':rnil':ok':nil'3(+(1, _n9278)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n9278)

Induction Base:
rcons'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)), _gen_mark':0':rnil':ok':nil'3(b))

Induction Step:
rcons'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n9279, 1))), _gen_mark':0':rnil':ok':nil'3(_b10855)) →_R^Ω(1)
mark'(rcons'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n9279)), _gen_mark':0':rnil':ok':nil'3(_b10855))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
posrecip', active', 2ndsneg', negrecip', 2ndspos', plus', times', pi', square', proper', top'

They will be analysed ascendingly in the following order:
posrecip' < active'
2ndsneg' < active'
negrecip' < active'
2ndspos' < active'
plus' < active'
times' < active'
pi' < active'
square' < active'
active' < top'
posrecip' < proper'
2ndsneg' < proper'
negrecip' < proper'
2ndspos' < proper'
plus' < proper'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Proved the following rewrite lemma:
posrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, _n14220))) → _*4, rt ∈ Ω(_n14220)

Induction Base:
posrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)))

Induction Step:
posrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n14221, 1)))) →_R^Ω(1)
mark'(posrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n14221)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
2ndsneg', active', negrecip', 2ndspos', plus', times', pi', square', proper', top'

They will be analysed ascendingly in the following order:
2ndsneg' < active'
negrecip' < active'
2ndspos' < active'
plus' < active'
times' < active'
pi' < active'
square' < active'
active' < top'
2ndsneg' < proper'
negrecip' < proper'
2ndspos' < proper'
plus' < proper'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Proved the following rewrite lemma:
2ndsneg'(_gen_mark':0':rnil':ok':nil'3(+(1, _n17297)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n17297)

Induction Base:
2ndsneg'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)), _gen_mark':0':rnil':ok':nil'3(b))

Induction Step:
2ndsneg'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n17298, 1))), _gen_mark':0':rnil':ok':nil'3(_b19414)) →_R^Ω(1)
mark'(2ndsneg'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n17298)), _gen_mark':0':rnil':ok':nil'3(_b19414))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
negrecip', active', 2ndspos', plus', times', pi', square', proper', top'

They will be analysed ascendingly in the following order:
negrecip' < active'
2ndspos' < active'
plus' < active'
times' < active'
pi' < active'
square' < active'
active' < top'
negrecip' < proper'
2ndspos' < proper'
plus' < proper'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Proved the following rewrite lemma:
negrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, _n22848))) → _*4, rt ∈ Ω(_n22848)

Induction Base:
negrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)))

Induction Step:
negrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n22849, 1)))) →_R^Ω(1)
mark'(negrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n22849)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
2ndspos', active', plus', times', pi', square', proper', top'

They will be analysed ascendingly in the following order:
2ndspos' < active'
plus' < active'
times' < active'
pi' < active'
square' < active'
active' < top'
2ndspos' < proper'
plus' < proper'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Proved the following rewrite lemma:
2ndspos'(_gen_mark':0':rnil':ok':nil'3(+(1, _n26240)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n26240)

Induction Base:
2ndspos'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)), _gen_mark':0':rnil':ok':nil'3(b))

Induction Step:
2ndspos'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n26241, 1))), _gen_mark':0':rnil':ok':nil'3(_b28897)) →_R^Ω(1)
mark'(2ndspos'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n26241)), _gen_mark':0':rnil':ok':nil'3(_b28897))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
plus', active', times', pi', square', proper', top'

They will be analysed ascendingly in the following order:
plus' < active'
times' < active'
pi' < active'
square' < active'
active' < top'
plus' < proper'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Proved the following rewrite lemma:
plus'(_gen_mark':0':rnil':ok':nil'3(+(1, _n32400)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n32400)

Induction Base:
plus'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)), _gen_mark':0':rnil':ok':nil'3(b))

Induction Step:
plus'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n32401, 1))), _gen_mark':0':rnil':ok':nil'3(_b35381)) →_R^Ω(1)
mark'(plus'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n32401)), _gen_mark':0':rnil':ok':nil'3(_b35381))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_mark':0':rnil':ok':nil'3(+(1, _n5)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n5)
from'(_gen_mark':0':rnil':ok':nil'3(+(1, _n3878))) → _*4, rt ∈ Ω(_n3878)
s'(_gen_mark':0':rnil':ok':nil'3(+(1, _n6516))) → _*4, rt ∈ Ω(_n6516)
rcons'(_gen_mark':0':rnil':ok':nil'3(+(1, _n9278)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n9278)
posrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, _n14220))) → _*4, rt ∈ Ω(_n14220)
2ndsneg'(_gen_mark':0':rnil':ok':nil'3(+(1, _n17297)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n17297)
negrecip'(_gen_mark':0':rnil':ok':nil'3(+(1, _n22848))) → _*4, rt ∈ Ω(_n22848)
2ndspos'(_gen_mark':0':rnil':ok':nil'3(+(1, _n26240)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n26240)
plus'(_gen_mark':0':rnil':ok':nil'3(+(1, _n32400)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n32400)

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
times', active', pi', square', proper', top'

They will be analysed ascendingly in the following order:
times' < active'
pi' < active'
square' < active'
active' < top'
times' < proper'
pi' < proper'
square' < proper'
proper' < top'

Proved the following rewrite lemma:
times'(_gen_mark':0':rnil':ok':nil'3(+(1, _n38928)), _gen_mark':0':rnil':ok':nil'3(b)) → _*4, rt ∈ Ω(_n38928)

Induction Base:
times'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)), _gen_mark':0':rnil':ok':nil'3(b))

Induction Step:
times'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n38929, 1))), _gen_mark':0':rnil':ok':nil'3(_b42233)) →_R^Ω(1)
mark'(times'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n38929)), _gen_mark':0':rnil':ok':nil'3(_b42233))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

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

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

Proved the following rewrite lemma:
pi'(_gen_mark':0':rnil':ok':nil'3(+(1, _n45824))) → _*4, rt ∈ Ω(_n45824)

Induction Base:
pi'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)))

Induction Step:
pi'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n45825, 1)))) →_R^Ω(1)
mark'(pi'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n45825)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

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

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

Proved the following rewrite lemma:
square'(_gen_mark':0':rnil':ok':nil'3(+(1, _n49913))) → _*4, rt ∈ Ω(_n49913)

Induction Base:
square'(_gen_mark':0':rnil':ok':nil'3(+(1, 0)))

Induction Step:
square'(_gen_mark':0':rnil':ok':nil'3(+(1, +(_$n49914, 1)))) →_R^Ω(1)
mark'(square'(_gen_mark':0':rnil':ok':nil'3(+(1, _$n49914)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(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'

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

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

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

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

Generator Equations:
_gen_mark':0':rnil':ok':nil'3(0) ⇔ 0'
_gen_mark':0':rnil':ok':nil'3(+(x, 1)) ⇔ mark'(_gen_mark':0':rnil':ok':nil'3(x))

The following defined symbols remain to be analysed:
top'