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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(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'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Heuristically decided to analyse the following defined symbols:
active', sel', cons', first', from', s', sel1', quote', cons1', first1', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
sel' < active'
cons' < active'
first' < active'
from' < active'
s' < active'
sel1' < active'
quote' < active'
cons1' < active'
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
sel' < proper'
cons' < proper'
first' < proper'
from' < proper'
s' < proper'
sel1' < proper'
quote' < proper'
cons1' < proper'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
sel', active', cons', first', from', s', sel1', quote', cons1', first1', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
sel' < active'
cons' < active'
first' < active'
from' < active'
s' < active'
sel1' < active'
quote' < active'
cons1' < active'
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
sel' < proper'
cons' < proper'
first' < proper'
from' < proper'
s' < proper'
sel1' < proper'
quote' < proper'
cons1' < proper'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)

Induction Base:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)), _gen_mark':0':nil':nil1':01':ok'3(b))

Induction Step:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n6, 1))), _gen_mark':0':nil':nil1':01':ok'3(_b826)) →RΩ(1)
mark'(sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n6)), _gen_mark':0':nil':nil1':01':ok'3(_b826))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
cons', active', first', from', s', sel1', quote', cons1', first1', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
first' < active'
from' < active'
s' < active'
sel1' < active'
quote' < active'
cons1' < active'
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
cons' < proper'
first' < proper'
from' < proper'
s' < proper'
sel1' < proper'
quote' < proper'
cons1' < proper'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)

Induction Base:
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)), _gen_mark':0':nil':nil1':01':ok'3(b))

Induction Step:
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n4794, 1))), _gen_mark':0':nil':nil1':01':ok'3(_b5722)) →RΩ(1)
mark'(cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n4794)), _gen_mark':0':nil':nil1':01':ok'3(_b5722))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
first', active', from', s', sel1', quote', cons1', first1', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
first' < active'
from' < active'
s' < active'
sel1' < active'
quote' < active'
cons1' < active'
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
first' < proper'
from' < proper'
s' < proper'
sel1' < proper'
quote' < proper'
cons1' < proper'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)

Induction Base:
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)), _gen_mark':0':nil':nil1':01':ok'3(b))

Induction Step:
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n9731, 1))), _gen_mark':0':nil':nil1':01':ok'3(_b11199)) →RΩ(1)
mark'(first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n9731)), _gen_mark':0':nil':nil1':01':ok'3(_b11199))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
from', active', s', sel1', quote', cons1', first1', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
from' < active'
s' < active'
sel1' < active'
quote' < active'
cons1' < active'
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
from' < proper'
s' < proper'
sel1' < proper'
quote' < proper'
cons1' < proper'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)

Induction Base:
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)))

Induction Step:
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n15255, 1)))) →RΩ(1)
mark'(from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n15255)))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
s', active', sel1', quote', cons1', first1', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
sel1' < active'
quote' < active'
cons1' < active'
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
s' < proper'
sel1' < proper'
quote' < proper'
cons1' < proper'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)

Induction Base:
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)))

Induction Step:
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n18731, 1)))) →RΩ(1)
mark'(s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n18731)))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
sel1', active', quote', cons1', first1', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
sel1' < active'
quote' < active'
cons1' < active'
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
sel1' < proper'
quote' < proper'
cons1' < proper'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)

Induction Base:
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)), _gen_mark':0':nil':nil1':01':ok'3(b))

Induction Step:
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n22331, 1))), _gen_mark':0':nil':nil1':01':ok'3(_b24555)) →RΩ(1)
mark'(sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n22331)), _gen_mark':0':nil':nil1':01':ok'3(_b24555))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
quote', active', cons1', first1', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
quote' < active'
cons1' < active'
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
quote' < proper'
cons1' < proper'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
cons1', active', first1', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
cons1' < active'
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
cons1' < proper'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n28762)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n28762)

Induction Base:
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)), _gen_mark':0':nil':nil1':01':ok'3(b))

Induction Step:
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n28763, 1))), _gen_mark':0':nil':nil1':01':ok'3(_b31311)) →RΩ(1)
mark'(cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n28763)), _gen_mark':0':nil':nil1':01':ok'3(_b31311))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n28762)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n28762)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
first1', active', quote1', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
first1' < active'
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
first1' < proper'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n35504)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n35504)

Induction Base:
first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)), _gen_mark':0':nil':nil1':01':ok'3(b))

Induction Step:
first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n35505, 1))), _gen_mark':0':nil':nil1':01':ok'3(_b38377)) →RΩ(1)
mark'(first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n35505)), _gen_mark':0':nil':nil1':01':ok'3(_b38377))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n28762)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n28762)
first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n35504)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n35504)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
quote1', active', s1', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
quote1' < active'
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
quote1' < proper'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Could not prove a rewrite lemma for the defined symbol quote1'.

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n28762)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n28762)
first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n35504)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n35504)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
s1', active', unquote', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
s1' < active'
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
s1' < proper'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
s1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n42692))) → _*4, rt ∈ Ω(n42692)

Induction Base:
s1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)))

Induction Step:
s1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n42693, 1)))) →RΩ(1)
mark'(s1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n42693)))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n28762)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n28762)
first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n35504)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n35504)
s1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n42692))) → _*4, rt ∈ Ω(n42692)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
unquote', active', fcons', unquote1', proper', top'

They will be analysed ascendingly in the following order:
unquote' < active'
fcons' < active'
unquote1' < active'
active' < top'
unquote' < proper'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
unquote'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n46989))) → _*4, rt ∈ Ω(n46989)

Induction Base:
unquote'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)))

Induction Step:
unquote'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n46990, 1)))) →RΩ(1)
mark'(unquote'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n46990)))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n28762)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n28762)
first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n35504)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n35504)
s1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n42692))) → _*4, rt ∈ Ω(n42692)
unquote'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n46989))) → _*4, rt ∈ Ω(n46989)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
fcons', active', unquote1', proper', top'

They will be analysed ascendingly in the following order:
fcons' < active'
unquote1' < active'
active' < top'
fcons' < proper'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
fcons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n51410)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n51410)

Induction Base:
fcons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)), _gen_mark':0':nil':nil1':01':ok'3(b))

Induction Step:
fcons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n51411, 1))), _gen_mark':0':nil':nil1':01':ok'3(_b55039)) →RΩ(1)
mark'(fcons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n51411)), _gen_mark':0':nil':nil1':01':ok'3(_b55039))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n28762)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n28762)
first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n35504)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n35504)
s1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n42692))) → _*4, rt ∈ Ω(n42692)
unquote'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n46989))) → _*4, rt ∈ Ω(n46989)
fcons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n51410)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n51410)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':ok'3(x))

The following defined symbols remain to be analysed:
unquote1', active', proper', top'

They will be analysed ascendingly in the following order:
unquote1' < active'
active' < top'
unquote1' < proper'
proper' < top'

Proved the following rewrite lemma:
unquote1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n59370))) → _*4, rt ∈ Ω(n59370)

Induction Base:
unquote1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, 0)))

Induction Step:
unquote1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, +(_\$n59371, 1)))) →RΩ(1)
mark'(unquote1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _\$n59371)))) →IH
mark'(_*4)

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

Rules:
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(sel'(0', cons'(X, Z))) → mark'(X)
active'(first'(0', Z)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel1'(s'(X), cons'(Y, Z))) → mark'(sel1'(X, Z))
active'(sel1'(0', cons'(X, Z))) → mark'(quote'(X))
active'(first1'(0', Z)) → mark'(nil1')
active'(first1'(s'(X), cons'(Y, Z))) → mark'(cons1'(quote'(Y), first1'(X, Z)))
active'(quote'(0')) → mark'(01')
active'(quote1'(cons'(X, Z))) → mark'(cons1'(quote'(X), quote1'(Z)))
active'(quote1'(nil')) → mark'(nil1')
active'(quote'(s'(X))) → mark'(s1'(quote'(X)))
active'(quote'(sel'(X, Z))) → mark'(sel1'(X, Z))
active'(quote1'(first'(X, Z))) → mark'(first1'(X, Z))
active'(unquote'(01')) → mark'(0')
active'(unquote'(s1'(X))) → mark'(s'(unquote'(X)))
active'(unquote1'(nil1')) → mark'(nil')
active'(unquote1'(cons1'(X, Z))) → mark'(fcons'(unquote'(X), unquote1'(Z)))
active'(fcons'(X, Z)) → mark'(cons'(X, Z))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(sel1'(X1, X2)) → sel1'(active'(X1), X2)
active'(sel1'(X1, X2)) → sel1'(X1, active'(X2))
active'(first1'(X1, X2)) → first1'(active'(X1), X2)
active'(first1'(X1, X2)) → first1'(X1, active'(X2))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
active'(s1'(X)) → s1'(active'(X))
active'(unquote'(X)) → unquote'(active'(X))
active'(unquote1'(X)) → unquote1'(active'(X))
active'(fcons'(X1, X2)) → fcons'(active'(X1), X2)
active'(fcons'(X1, X2)) → fcons'(X1, active'(X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
sel1'(mark'(X1), X2) → mark'(sel1'(X1, X2))
sel1'(X1, mark'(X2)) → mark'(sel1'(X1, X2))
first1'(mark'(X1), X2) → mark'(first1'(X1, X2))
first1'(X1, mark'(X2)) → mark'(first1'(X1, X2))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
s1'(mark'(X)) → mark'(s1'(X))
unquote'(mark'(X)) → mark'(unquote'(X))
unquote1'(mark'(X)) → mark'(unquote1'(X))
fcons'(mark'(X1), X2) → mark'(fcons'(X1, X2))
fcons'(X1, mark'(X2)) → mark'(fcons'(X1, X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(from'(X)) → from'(proper'(X))
proper'(sel1'(X1, X2)) → sel1'(proper'(X1), proper'(X2))
proper'(quote'(X)) → quote'(proper'(X))
proper'(first1'(X1, X2)) → first1'(proper'(X1), proper'(X2))
proper'(nil1') → ok'(nil1')
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
proper'(01') → ok'(01')
proper'(quote1'(X)) → quote1'(proper'(X))
proper'(s1'(X)) → s1'(proper'(X))
proper'(unquote'(X)) → unquote'(proper'(X))
proper'(unquote1'(X)) → unquote1'(proper'(X))
proper'(fcons'(X1, X2)) → fcons'(proper'(X1), proper'(X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
sel1'(ok'(X1), ok'(X2)) → ok'(sel1'(X1, X2))
quote'(ok'(X)) → ok'(quote'(X))
first1'(ok'(X1), ok'(X2)) → ok'(first1'(X1, X2))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
quote1'(ok'(X)) → ok'(quote1'(X))
s1'(ok'(X)) → ok'(s1'(X))
unquote'(ok'(X)) → ok'(unquote'(X))
unquote1'(ok'(X)) → ok'(unquote1'(X))
fcons'(ok'(X1), ok'(X2)) → ok'(fcons'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
cons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
mark' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
0' :: mark':0':nil':nil1':01':ok'
first' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil' :: mark':0':nil':nil1':01':ok'
from' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
sel1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
quote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
first1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
nil1' :: mark':0':nil':nil1':01':ok'
cons1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
01' :: mark':0':nil':nil1':01':ok'
quote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
s1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
unquote1' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
fcons' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
proper' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
ok' :: mark':0':nil':nil1':01':ok' → mark':0':nil':nil1':01':ok'
top' :: mark':0':nil':nil1':01':ok' → top'
_hole_mark':0':nil':nil1':01':ok'1 :: mark':0':nil':nil1':01':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':nil1':01':ok'3 :: Nat → mark':0':nil':nil1':01':ok'

Lemmas:
sel'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n5)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n5)
cons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n4793)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n4793)
first'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n9730)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n9730)
from'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n15254))) → _*4, rt ∈ Ω(n15254)
s'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n18730))) → _*4, rt ∈ Ω(n18730)
sel1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n22330)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n22330)
cons1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n28762)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n28762)
first1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n35504)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n35504)
s1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n42692))) → _*4, rt ∈ Ω(n42692)
unquote'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n46989))) → _*4, rt ∈ Ω(n46989)
fcons'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n51410)), _gen_mark':0':nil':nil1':01':ok'3(b)) → _*4, rt ∈ Ω(n51410)
unquote1'(_gen_mark':0':nil':nil1':01':ok'3(+(1, _n59370))) → _*4, rt ∈ Ω(n59370)

Generator Equations:
_gen_mark':0':nil':nil1':01':ok'3(0) ⇔ 0'
_gen_mark':0':nil':nil1':01':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':nil1':01':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'