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

active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) → mark(0)
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0, X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(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'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Heuristically decided to analyse the following defined symbols:
active', cons', recip', sqr', terms', s', add', dbl', first', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
recip' < active'
sqr' < active'
terms' < active'
s' < active'
add' < active'
dbl' < active'
first' < active'
active' < top'
cons' < proper'
recip' < proper'
sqr' < proper'
terms' < proper'
s' < proper'
add' < proper'
dbl' < proper'
first' < proper'
proper' < top'

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

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

The following defined symbols remain to be analysed:
cons', active', recip', sqr', terms', s', add', dbl', first', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
recip' < active'
sqr' < active'
terms' < active'
s' < active'
add' < active'
dbl' < active'
first' < active'
active' < top'
cons' < proper'
recip' < proper'
sqr' < proper'
terms' < proper'
s' < proper'
add' < proper'
dbl' < proper'
first' < proper'
proper' < top'

Proved the following rewrite lemma:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)

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

Induction Step:
cons'(_gen_mark':0':nil':ok'3(+(1, +(_$n6, 1))), _gen_mark':0':nil':ok'3(_b610)) →RΩ(1)
mark'(cons'(_gen_mark':0':nil':ok'3(+(1, _$n6)), _gen_mark':0':nil':ok'3(_b610))) →IH
mark'(_*4)

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)

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

The following defined symbols remain to be analysed:
recip', active', sqr', terms', s', add', dbl', first', proper', top'

They will be analysed ascendingly in the following order:
recip' < active'
sqr' < active'
terms' < active'
s' < active'
add' < active'
dbl' < active'
first' < active'
active' < top'
recip' < proper'
sqr' < proper'
terms' < proper'
s' < proper'
add' < proper'
dbl' < proper'
first' < proper'
proper' < top'

Proved the following rewrite lemma:
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)

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

Induction Step:
recip'(_gen_mark':0':nil':ok'3(+(1, +(_$n2484, 1)))) →RΩ(1)
mark'(recip'(_gen_mark':0':nil':ok'3(+(1, _$n2484)))) →IH
mark'(_*4)

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)

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

The following defined symbols remain to be analysed:
sqr', active', terms', s', add', dbl', first', proper', top'

They will be analysed ascendingly in the following order:
sqr' < active'
terms' < active'
s' < active'
add' < active'
dbl' < active'
first' < active'
active' < top'
sqr' < proper'
terms' < proper'
s' < proper'
add' < proper'
dbl' < proper'
first' < proper'
proper' < top'

Proved the following rewrite lemma:
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)

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

Induction Step:
sqr'(_gen_mark':0':nil':ok'3(+(1, +(_$n4195, 1)))) →RΩ(1)
mark'(sqr'(_gen_mark':0':nil':ok'3(+(1, _$n4195)))) →IH
mark'(_*4)

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)

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

The following defined symbols remain to be analysed:
terms', active', s', add', dbl', first', proper', top'

They will be analysed ascendingly in the following order:
terms' < active'
s' < active'
add' < active'
dbl' < active'
first' < active'
active' < top'
terms' < proper'
s' < proper'
add' < proper'
dbl' < proper'
first' < proper'
proper' < top'

Proved the following rewrite lemma:
terms'(_gen_mark':0':nil':ok'3(+(1, _n6029))) → _*4, rt ∈ Ω(n6029)

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

Induction Step:
terms'(_gen_mark':0':nil':ok'3(+(1, +(_$n6030, 1)))) →RΩ(1)
mark'(terms'(_gen_mark':0':nil':ok'3(+(1, _$n6030)))) →IH
mark'(_*4)

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)
terms'(_gen_mark':0':nil':ok'3(+(1, _n6029))) → _*4, rt ∈ Ω(n6029)

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

The following defined symbols remain to be analysed:
s', active', add', dbl', first', proper', top'

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

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)
terms'(_gen_mark':0':nil':ok'3(+(1, _n6029))) → _*4, rt ∈ Ω(n6029)

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

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

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

Proved the following rewrite lemma:
add'(_gen_mark':0':nil':ok'3(+(1, _n8022)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n8022)

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

Induction Step:
add'(_gen_mark':0':nil':ok'3(+(1, +(_$n8023, 1))), _gen_mark':0':nil':ok'3(_b9815)) →RΩ(1)
mark'(add'(_gen_mark':0':nil':ok'3(+(1, _$n8023)), _gen_mark':0':nil':ok'3(_b9815))) →IH
mark'(_*4)

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)
terms'(_gen_mark':0':nil':ok'3(+(1, _n6029))) → _*4, rt ∈ Ω(n6029)
add'(_gen_mark':0':nil':ok'3(+(1, _n8022)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n8022)

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

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

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

Proved the following rewrite lemma:
dbl'(_gen_mark':0':nil':ok'3(+(1, _n11810))) → _*4, rt ∈ Ω(n11810)

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

Induction Step:
dbl'(_gen_mark':0':nil':ok'3(+(1, +(_$n11811, 1)))) →RΩ(1)
mark'(dbl'(_gen_mark':0':nil':ok'3(+(1, _$n11811)))) →IH
mark'(_*4)

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)
terms'(_gen_mark':0':nil':ok'3(+(1, _n6029))) → _*4, rt ∈ Ω(n6029)
add'(_gen_mark':0':nil':ok'3(+(1, _n8022)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n8022)
dbl'(_gen_mark':0':nil':ok'3(+(1, _n11810))) → _*4, rt ∈ Ω(n11810)

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

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

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

Proved the following rewrite lemma:
first'(_gen_mark':0':nil':ok'3(+(1, _n14084)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n14084)

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

Induction Step:
first'(_gen_mark':0':nil':ok'3(+(1, +(_$n14085, 1))), _gen_mark':0':nil':ok'3(_b16417)) →RΩ(1)
mark'(first'(_gen_mark':0':nil':ok'3(+(1, _$n14085)), _gen_mark':0':nil':ok'3(_b16417))) →IH
mark'(_*4)

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)
terms'(_gen_mark':0':nil':ok'3(+(1, _n6029))) → _*4, rt ∈ Ω(n6029)
add'(_gen_mark':0':nil':ok'3(+(1, _n8022)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n8022)
dbl'(_gen_mark':0':nil':ok'3(+(1, _n11810))) → _*4, rt ∈ Ω(n11810)
first'(_gen_mark':0':nil':ok'3(+(1, _n14084)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n14084)

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

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)
terms'(_gen_mark':0':nil':ok'3(+(1, _n6029))) → _*4, rt ∈ Ω(n6029)
add'(_gen_mark':0':nil':ok'3(+(1, _n8022)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n8022)
dbl'(_gen_mark':0':nil':ok'3(+(1, _n11810))) → _*4, rt ∈ Ω(n11810)
first'(_gen_mark':0':nil':ok'3(+(1, _n14084)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n14084)

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

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

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

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)
terms'(_gen_mark':0':nil':ok'3(+(1, _n6029))) → _*4, rt ∈ Ω(n6029)
add'(_gen_mark':0':nil':ok'3(+(1, _n8022)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n8022)
dbl'(_gen_mark':0':nil':ok'3(+(1, _n11810))) → _*4, rt ∈ Ω(n11810)
first'(_gen_mark':0':nil':ok'3(+(1, _n14084)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n14084)

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

The following defined symbols remain to be analysed:
top'

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

Rules:
active'(terms'(N)) → mark'(cons'(recip'(sqr'(N)), terms'(s'(N))))
active'(sqr'(0')) → mark'(0')
active'(sqr'(s'(X))) → mark'(s'(add'(sqr'(X), dbl'(X))))
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(terms'(X)) → terms'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(recip'(X)) → recip'(active'(X))
active'(sqr'(X)) → sqr'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(dbl'(X)) → dbl'(active'(X))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
terms'(mark'(X)) → mark'(terms'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
recip'(mark'(X)) → mark'(recip'(X))
sqr'(mark'(X)) → mark'(sqr'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
dbl'(mark'(X)) → mark'(dbl'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(terms'(X)) → terms'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(recip'(X)) → recip'(proper'(X))
proper'(sqr'(X)) → sqr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
terms'(ok'(X)) → ok'(terms'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
recip'(ok'(X)) → ok'(recip'(X))
sqr'(ok'(X)) → ok'(sqr'(X))
s'(ok'(X)) → ok'(s'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
dbl'(ok'(X)) → ok'(dbl'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
terms' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
recip' :: mark':0':nil':ok' → mark':0':nil':ok'
sqr' :: mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
add' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
dbl' :: mark':0':nil':ok' → mark':0':nil':ok'
first' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
recip'(_gen_mark':0':nil':ok'3(+(1, _n2483))) → _*4, rt ∈ Ω(n2483)
sqr'(_gen_mark':0':nil':ok'3(+(1, _n4194))) → _*4, rt ∈ Ω(n4194)
terms'(_gen_mark':0':nil':ok'3(+(1, _n6029))) → _*4, rt ∈ Ω(n6029)
add'(_gen_mark':0':nil':ok'3(+(1, _n8022)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n8022)
dbl'(_gen_mark':0':nil':ok'3(+(1, _n11810))) → _*4, rt ∈ Ω(n11810)
first'(_gen_mark':0':nil':ok'3(+(1, _n14084)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n14084)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)