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

(0) Obligation:

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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
s(mark(X)) →⁺ mark(s(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 [ ].

(2) BOUNDS(n^1, INF)

(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(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))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

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

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

(8) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

The following defined symbols remain to be analysed:
cons, active, 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

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

Proved the following rewrite lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))

Induction Step:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, +(n5_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →_R^Ω(1)
mark(cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b))) →_IH
mark(*4_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

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

Proved the following rewrite lemma:
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)

Induction Base:
from(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))

Induction Step:
from(gen_mark:0':rnil:ok:nil3_0(+(1, +(n1430_0, 1)))) →_R^Ω(1)
mark(from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0)))) →_IH
mark(*4_0)

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

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

Proved the following rewrite lemma:
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)

Induction Base:
s(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))

Induction Step:
s(gen_mark:0':rnil:ok:nil3_0(+(1, +(n2069_0, 1)))) →_R^Ω(1)
mark(s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0)))) →_IH
mark(*4_0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

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

Proved the following rewrite lemma:
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)

Induction Base:
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))

Induction Step:
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, +(n2809_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →_R^Ω(1)
mark(rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b))) →_IH
mark(*4_0)

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

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

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

Proved the following rewrite lemma:
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)

Induction Base:
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))

Induction Step:
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, +(n5149_0, 1)))) →_R^Ω(1)
mark(posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0)))) →_IH
mark(*4_0)

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

(22) Complex Obligation (BEST)

(23) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

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

Proved the following rewrite lemma:
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)

Induction Base:
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))

Induction Step:
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, +(n6140_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →_R^Ω(1)
mark(2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b))) →_IH
mark(*4_0)

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

(25) Complex Obligation (BEST)

(26) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

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

Proved the following rewrite lemma:
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)

Induction Base:
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))

Induction Step:
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, +(n8990_0, 1)))) →_R^Ω(1)
mark(negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0)))) →_IH
mark(*4_0)

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

(28) Complex Obligation (BEST)

(29) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

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

Proved the following rewrite lemma:
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)

Induction Base:
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))

Induction Step:
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, +(n10232_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →_R^Ω(1)
mark(2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b))) →_IH
mark(*4_0)

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

(31) Complex Obligation (BEST)

(32) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

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

Proved the following rewrite lemma:
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)

Induction Base:
plus(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))

Induction Step:
plus(gen_mark:0':rnil:ok:nil3_0(+(1, +(n13592_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →_R^Ω(1)
mark(plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b))) →_IH
mark(*4_0)

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

(34) Complex Obligation (BEST)

(35) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

(36) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)

Induction Base:
times(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))

Induction Step:
times(gen_mark:0':rnil:ok:nil3_0(+(1, +(n17256_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →_R^Ω(1)
mark(times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b))) →_IH
mark(*4_0)

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

(37) Complex Obligation (BEST)

(38) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

(39) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n21224_0))) → *4_0, rt ∈ Ω(n21224₀)

Induction Base:
pi(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))

Induction Step:
pi(gen_mark:0':rnil:ok:nil3_0(+(1, +(n21224_0, 1)))) →_R^Ω(1)
mark(pi(gen_mark:0':rnil:ok:nil3_0(+(1, n21224_0)))) →_IH
mark(*4_0)

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

(40) Complex Obligation (BEST)

(41) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n21224_0))) → *4_0, rt ∈ Ω(n21224₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(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

(42) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
square(gen_mark:0':rnil:ok:nil3_0(+(1, n23017_0))) → *4_0, rt ∈ Ω(n23017₀)

Induction Base:
square(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))

Induction Step:
square(gen_mark:0':rnil:ok:nil3_0(+(1, +(n23017_0, 1)))) →_R^Ω(1)
mark(square(gen_mark:0':rnil:ok:nil3_0(+(1, n23017_0)))) →_IH
mark(*4_0)

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

(43) Complex Obligation (BEST)

(44) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n21224_0))) → *4_0, rt ∈ Ω(n21224₀)
square(gen_mark:0':rnil:ok:nil3_0(+(1, n23017_0))) → *4_0, rt ∈ Ω(n23017₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_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

(45) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(46) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n21224_0))) → *4_0, rt ∈ Ω(n21224₀)
square(gen_mark:0':rnil:ok:nil3_0(+(1, n23017_0))) → *4_0, rt ∈ Ω(n23017₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

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

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

(47) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(48) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n21224_0))) → *4_0, rt ∈ Ω(n21224₀)
square(gen_mark:0':rnil:ok:nil3_0(+(1, n23017_0))) → *4_0, rt ∈ Ω(n23017₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

The following defined symbols remain to be analysed:
top

(49) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(50) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n21224_0))) → *4_0, rt ∈ Ω(n21224₀)
square(gen_mark:0':rnil:ok:nil3_0(+(1, n23017_0))) → *4_0, rt ∈ Ω(n23017₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(51) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(52) BOUNDS(n^1, INF)

(53) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n21224_0))) → *4_0, rt ∈ Ω(n21224₀)
square(gen_mark:0':rnil:ok:nil3_0(+(1, n23017_0))) → *4_0, rt ∈ Ω(n23017₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(54) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(55) BOUNDS(n^1, INF)

(56) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n21224_0))) → *4_0, rt ∈ Ω(n21224₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(57) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(58) BOUNDS(n^1, INF)

(59) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n17256_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n17256₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(60) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(61) BOUNDS(n^1, INF)

(62) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n13592_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n13592₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(63) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(64) BOUNDS(n^1, INF)

(65) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n10232_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n10232₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(66) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(67) BOUNDS(n^1, INF)

(68) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n8990_0))) → *4_0, rt ∈ Ω(n8990₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(69) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(70) BOUNDS(n^1, INF)

(71) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n6140_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n6140₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(72) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(73) BOUNDS(n^1, INF)

(74) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n5149_0))) → *4_0, rt ∈ Ω(n5149₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(75) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(76) BOUNDS(n^1, INF)

(77) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n2809_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n2809₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(78) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(79) BOUNDS(n^1, INF)

(80) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2069_0))) → *4_0, rt ∈ Ω(n2069₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(81) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(82) BOUNDS(n^1, INF)

(83) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1430_0))) → *4_0, rt ∈ Ω(n1430₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(84) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(85) BOUNDS(n^1, INF)

(86) Obligation:

TRS:
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:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil

Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))

No more defined symbols left to analyse.

(87) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(88) BOUNDS(n^1, INF)