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

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

active'(and'(tt', T)) → mark'(T)
active'(isNatIList'(IL)) → mark'(isNatList'(IL))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(s'(N))) → mark'(isNat'(N))
active'(isNat'(length'(L))) → mark'(isNatList'(L))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(N, IL))) → mark'(and'(isNat'(N), isNatIList'(IL)))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(N, L))) → mark'(and'(isNat'(N), isNatList'(L)))
active'(isNatList'(take'(N, IL))) → mark'(and'(isNat'(N), isNatIList'(IL)))
active'(zeros') → mark'(cons'(0', zeros'))
active'(take'(0', IL)) → mark'(uTake1'(isNatIList'(IL)))
active'(uTake1'(tt')) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(uTake2'(and'(isNat'(M), and'(isNat'(N), isNatIList'(IL))), M, N, IL))
active'(uTake2'(tt', M, N, IL)) → mark'(cons'(N, take'(M, IL)))
active'(length'(cons'(N, L))) → mark'(uLength'(and'(isNat'(N), isNatList'(L)), L))
active'(uLength'(tt', L)) → mark'(s'(length'(L)))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(and'(X1, X2)) → and'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(uTake1'(X)) → uTake1'(active'(X))
active'(uTake2'(X1, X2, X3, X4)) → uTake2'(active'(X1), X2, X3, X4)
active'(uLength'(X1, X2)) → uLength'(active'(X1), X2)
and'(mark'(X1), X2) → mark'(and'(X1, X2))
and'(X1, mark'(X2)) → mark'(and'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
uTake1'(mark'(X)) → mark'(uTake1'(X))
uTake2'(mark'(X1), X2, X3, X4) → mark'(uTake2'(X1, X2, X3, X4))
uLength'(mark'(X1), X2) → mark'(uLength'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(uTake1'(X)) → uTake1'(proper'(X))
proper'(uTake2'(X1, X2, X3, X4)) → uTake2'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(uLength'(X1, X2)) → uLength'(proper'(X1), proper'(X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
isNat'(ok'(X)) → ok'(isNat'(X))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
uTake1'(ok'(X)) → ok'(uTake1'(X))
uTake2'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(uTake2'(X1, X2, X3, X4))
uLength'(ok'(X1), ok'(X2)) → ok'(uLength'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(and'(tt', T)) → mark'(T)
active'(isNatIList'(IL)) → mark'(isNatList'(IL))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(s'(N))) → mark'(isNat'(N))
active'(isNat'(length'(L))) → mark'(isNatList'(L))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(N, IL))) → mark'(and'(isNat'(N), isNatIList'(IL)))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(N, L))) → mark'(and'(isNat'(N), isNatList'(L)))
active'(isNatList'(take'(N, IL))) → mark'(and'(isNat'(N), isNatIList'(IL)))
active'(zeros') → mark'(cons'(0', zeros'))
active'(take'(0', IL)) → mark'(uTake1'(isNatIList'(IL)))
active'(uTake1'(tt')) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(uTake2'(and'(isNat'(M), and'(isNat'(N), isNatIList'(IL))), M, N, IL))
active'(uTake2'(tt', M, N, IL)) → mark'(cons'(N, take'(M, IL)))
active'(length'(cons'(N, L))) → mark'(uLength'(and'(isNat'(N), isNatList'(L)), L))
active'(uLength'(tt', L)) → mark'(s'(length'(L)))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(and'(X1, X2)) → and'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(uTake1'(X)) → uTake1'(active'(X))
active'(uTake2'(X1, X2, X3, X4)) → uTake2'(active'(X1), X2, X3, X4)
active'(uLength'(X1, X2)) → uLength'(active'(X1), X2)
and'(mark'(X1), X2) → mark'(and'(X1, X2))
and'(X1, mark'(X2)) → mark'(and'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
uTake1'(mark'(X)) → mark'(uTake1'(X))
uTake2'(mark'(X1), X2, X3, X4) → mark'(uTake2'(X1, X2, X3, X4))
uLength'(mark'(X1), X2) → mark'(uLength'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(uTake1'(X)) → uTake1'(proper'(X))
proper'(uTake2'(X1, X2, X3, X4)) → uTake2'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(uLength'(X1, X2)) → uLength'(proper'(X1), proper'(X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
isNat'(ok'(X)) → ok'(isNat'(X))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
uTake1'(ok'(X)) → ok'(uTake1'(X))
uTake2'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(uTake2'(X1, X2, X3, X4))
uLength'(ok'(X1), ok'(X2)) → ok'(uLength'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
and' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
tt' :: tt':mark':0':zeros':nil':ok'
mark' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
isNatIList' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
isNatList' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
isNat' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
0' :: tt':mark':0':zeros':nil':ok'
s' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
length' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
zeros' :: tt':mark':0':zeros':nil':ok'
cons' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
nil' :: tt':mark':0':zeros':nil':ok'
take' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
uTake1' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
uTake2' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
uLength' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
proper' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
ok' :: tt':mark':0':zeros':nil':ok' → tt':mark':0':zeros':nil':ok'
top' :: tt':mark':0':zeros':nil':ok' → top'
_hole_tt':mark':0':zeros':nil':ok'1 :: tt':mark':0':zeros':nil':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':zeros':nil':ok'3 :: Nat → tt':mark':0':zeros':nil':ok'

Heuristically decided to analyse the following defined symbols:
active', isNatList', isNat', and', isNatIList', cons', uTake1', uTake2', take', uLength', s', length', proper', top'

They will be analysed ascendingly in the following order:
isNatList' < active'
isNat' < active'
and' < active'
isNatIList' < active'
cons' < active'
uTake1' < active'
uTake2' < active'
take' < active'
uLength' < active'
s' < active'
length' < active'
active' < top'
isNatList' < proper'
isNat' < proper'
and' < proper'
isNatIList' < proper'
cons' < proper'
uTake1' < proper'
uTake2' < proper'
take' < proper'
uLength' < proper'
s' < proper'
length' < proper'
proper' < top'

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

The following defined symbols remain to be analysed:
isNatList', active', isNat', and', isNatIList', cons', uTake1', uTake2', take', uLength', s', length', proper', top'

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

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

The following defined symbols remain to be analysed:
isNat', active', and', isNatIList', cons', uTake1', uTake2', take', uLength', s', length', proper', top'

They will be analysed ascendingly in the following order:
isNat' < active'
and' < active'
isNatIList' < active'
cons' < active'
uTake1' < active'
uTake2' < active'
take' < active'
uLength' < active'
s' < active'
length' < active'
active' < top'
isNat' < proper'
and' < proper'
isNatIList' < proper'
cons' < proper'
uTake1' < proper'
uTake2' < proper'
take' < proper'
uLength' < proper'
s' < proper'
length' < proper'
proper' < top'

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

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

The following defined symbols remain to be analysed:
and', active', isNatIList', cons', uTake1', uTake2', take', uLength', s', length', proper', top'

They will be analysed ascendingly in the following order:
and' < active'
isNatIList' < active'
cons' < active'
uTake1' < active'
uTake2' < active'
take' < active'
uLength' < active'
s' < active'
length' < active'
active' < top'
and' < proper'
isNatIList' < proper'
cons' < proper'
uTake1' < proper'
uTake2' < proper'
take' < proper'
uLength' < proper'
s' < proper'
length' < proper'
proper' < top'

Proved the following rewrite lemma:
and'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n17)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n17)

Induction Base:
and'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, 0)), _gen_tt':mark':0':zeros':nil':ok'3(b))

Induction Step:
and'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, +(_$n18, 1))), _gen_tt':mark':0':zeros':nil':ok'3(_b838)) →_R^Ω(1)
mark'(and'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _$n18)), _gen_tt':mark':0':zeros':nil':ok'3(_b838))) →_IH
mark'(_*4)

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

Lemmas:
and'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n17)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n17)

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

The following defined symbols remain to be analysed:
isNatIList', active', cons', uTake1', uTake2', take', uLength', s', length', proper', top'

They will be analysed ascendingly in the following order:
isNatIList' < active'
cons' < active'
uTake1' < active'
uTake2' < active'
take' < active'
uLength' < active'
s' < active'
length' < active'
active' < top'
isNatIList' < proper'
cons' < proper'
uTake1' < proper'
uTake2' < proper'
take' < proper'
uLength' < proper'
s' < proper'
length' < proper'
proper' < top'

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

Lemmas:
and'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n17)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n17)

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

The following defined symbols remain to be analysed:
cons', active', uTake1', uTake2', take', uLength', s', length', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
uTake1' < active'
uTake2' < active'
take' < active'
uLength' < active'
s' < active'
length' < active'
active' < top'
cons' < proper'
uTake1' < proper'
uTake2' < proper'
take' < proper'
uLength' < proper'
s' < proper'
length' < proper'
proper' < top'

Proved the following rewrite lemma:
cons'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n3660)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n3660)

Induction Base:
cons'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, 0)), _gen_tt':mark':0':zeros':nil':ok'3(b))

Induction Step:
cons'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, +(_$n3661, 1))), _gen_tt':mark':0':zeros':nil':ok'3(_b4589)) →_R^Ω(1)
mark'(cons'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _$n3661)), _gen_tt':mark':0':zeros':nil':ok'3(_b4589))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

The following defined symbols remain to be analysed:
uTake1', active', uTake2', take', uLength', s', length', proper', top'

They will be analysed ascendingly in the following order:
uTake1' < active'
uTake2' < active'
take' < active'
uLength' < active'
s' < active'
length' < active'
active' < top'
uTake1' < proper'
uTake2' < proper'
take' < proper'
uLength' < proper'
s' < proper'
length' < proper'
proper' < top'

Proved the following rewrite lemma:
uTake1'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n7436))) → _*4, rt ∈ Ω(_n7436)

Induction Base:
uTake1'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, 0)))

Induction Step:
uTake1'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, +(_$n7437, 1)))) →_R^Ω(1)
mark'(uTake1'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _$n7437)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

The following defined symbols remain to be analysed:
uTake2', active', take', uLength', s', length', proper', top'

They will be analysed ascendingly in the following order:
uTake2' < active'
take' < active'
uLength' < active'
s' < active'
length' < active'
active' < top'
uTake2' < proper'
take' < proper'
uLength' < proper'
s' < proper'
length' < proper'
proper' < top'

Proved the following rewrite lemma:
uTake2'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n9956)), _gen_tt':mark':0':zeros':nil':ok'3(b), _gen_tt':mark':0':zeros':nil':ok'3(c), _gen_tt':mark':0':zeros':nil':ok'3(d)) → _*4, rt ∈ Ω(_n9956)

Induction Base:
uTake2'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, 0)), _gen_tt':mark':0':zeros':nil':ok'3(b), _gen_tt':mark':0':zeros':nil':ok'3(c), _gen_tt':mark':0':zeros':nil':ok'3(d))

Induction Step:
uTake2'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, +(_$n9957, 1))), _gen_tt':mark':0':zeros':nil':ok'3(_b13321), _gen_tt':mark':0':zeros':nil':ok'3(_c13322), _gen_tt':mark':0':zeros':nil':ok'3(_d13323)) →_R^Ω(1)
mark'(uTake2'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _$n9957)), _gen_tt':mark':0':zeros':nil':ok'3(_b13321), _gen_tt':mark':0':zeros':nil':ok'3(_c13322), _gen_tt':mark':0':zeros':nil':ok'3(_d13323))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

The following defined symbols remain to be analysed:
take', active', uLength', s', length', proper', top'

They will be analysed ascendingly in the following order:
take' < active'
uLength' < active'
s' < active'
length' < active'
active' < top'
take' < proper'
uLength' < proper'
s' < proper'
length' < proper'
proper' < top'

Proved the following rewrite lemma:
take'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n18279)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n18279)

Induction Base:
take'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, 0)), _gen_tt':mark':0':zeros':nil':ok'3(b))

Induction Step:
take'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, +(_$n18280, 1))), _gen_tt':mark':0':zeros':nil':ok'3(_b20504)) →_R^Ω(1)
mark'(take'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _$n18280)), _gen_tt':mark':0':zeros':nil':ok'3(_b20504))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

The following defined symbols remain to be analysed:
uLength', active', s', length', proper', top'

They will be analysed ascendingly in the following order:
uLength' < active'
s' < active'
length' < active'
active' < top'
uLength' < proper'
s' < proper'
length' < proper'
proper' < top'

Proved the following rewrite lemma:
uLength'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n23505)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n23505)

Induction Base:
uLength'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, 0)), _gen_tt':mark':0':zeros':nil':ok'3(b))

Induction Step:
uLength'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, +(_$n23506, 1))), _gen_tt':mark':0':zeros':nil':ok'3(_b25838)) →_R^Ω(1)
mark'(uLength'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _$n23506)), _gen_tt':mark':0':zeros':nil':ok'3(_b25838))) →_IH
mark'(_*4)

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

Lemmas:
and'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n17)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n17)
cons'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n3660)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n3660)
uTake1'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n7436))) → _*4, rt ∈ Ω(_n7436)
uTake2'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n9956)), _gen_tt':mark':0':zeros':nil':ok'3(b), _gen_tt':mark':0':zeros':nil':ok'3(c), _gen_tt':mark':0':zeros':nil':ok'3(d)) → _*4, rt ∈ Ω(_n9956)
take'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n18279)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n18279)
uLength'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n23505)), _gen_tt':mark':0':zeros':nil':ok'3(b)) → _*4, rt ∈ Ω(_n23505)

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

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

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

Proved the following rewrite lemma:
s'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n28880))) → _*4, rt ∈ Ω(_n28880)

Induction Base:
s'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, 0)))

Induction Step:
s'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, +(_$n28881, 1)))) →_R^Ω(1)
mark'(s'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _$n28881)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

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

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

Proved the following rewrite lemma:
length'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _n32231))) → _*4, rt ∈ Ω(_n32231)

Induction Base:
length'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, 0)))

Induction Step:
length'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, +(_$n32232, 1)))) →_R^Ω(1)
mark'(length'(_gen_tt':mark':0':zeros':nil':ok'3(+(1, _$n32232)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':zeros':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':zeros':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':zeros':nil':ok'3(x))

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

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