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

active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0, X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(add(X1, X2)) → add(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
add(mark(X1), X2) → mark(add(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Heuristically decided to analyse the following defined symbols:
active', s', add', cons', first', from', and', if', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
add' < active'
cons' < active'
first' < active'
from' < active'
and' < active'
if' < active'
active' < top'
s' < proper'
add' < proper'
cons' < proper'
first' < proper'
from' < proper'
and' < proper'
if' < proper'
proper' < top'

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

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

The following defined symbols remain to be analysed:
s', active', add', cons', first', from', and', if', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
add' < active'
cons' < active'
first' < active'
from' < active'
and' < active'
if' < active'
active' < top'
s' < proper'
add' < proper'
cons' < proper'
first' < proper'
from' < proper'
and' < proper'
if' < proper'
proper' < top'

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

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

The following defined symbols remain to be analysed:
add', active', cons', first', from', and', if', proper', top'

They will be analysed ascendingly in the following order:
add' < active'
cons' < active'
first' < active'
from' < active'
and' < active'
if' < active'
active' < top'
add' < proper'
cons' < proper'
first' < proper'
from' < proper'
and' < proper'
if' < proper'
proper' < top'

Proved the following rewrite lemma:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)

Induction Base:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, 0)), _gen_true':mark':false':0':nil':ok'3(b))

Induction Step:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, +(_$n12, 1))), _gen_true':mark':false':0':nil':ok'3(_b616)) →RΩ(1)
mark'(add'(_gen_true':mark':false':0':nil':ok'3(+(1, _$n12)), _gen_true':mark':false':0':nil':ok'3(_b616))) →IH
mark'(_*4)

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Lemmas:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)

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

The following defined symbols remain to be analysed:
cons', active', first', from', and', if', proper', top'

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

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Lemmas:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)

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

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

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

Proved the following rewrite lemma:
first'(_gen_true':mark':false':0':nil':ok'3(+(1, _n2463)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n2463)

Induction Base:
first'(_gen_true':mark':false':0':nil':ok'3(+(1, 0)), _gen_true':mark':false':0':nil':ok'3(b))

Induction Step:
first'(_gen_true':mark':false':0':nil':ok'3(+(1, +(_$n2464, 1))), _gen_true':mark':false':0':nil':ok'3(_b3608)) →RΩ(1)
mark'(first'(_gen_true':mark':false':0':nil':ok'3(+(1, _$n2464)), _gen_true':mark':false':0':nil':ok'3(_b3608))) →IH
mark'(_*4)

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Lemmas:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)
first'(_gen_true':mark':false':0':nil':ok'3(+(1, _n2463)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n2463)

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

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

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

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Lemmas:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)
first'(_gen_true':mark':false':0':nil':ok'3(+(1, _n2463)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n2463)

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

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

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

Proved the following rewrite lemma:
and'(_gen_true':mark':false':0':nil':ok'3(+(1, _n5500)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5500)

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

Induction Step:
and'(_gen_true':mark':false':0':nil':ok'3(+(1, +(_$n5501, 1))), _gen_true':mark':false':0':nil':ok'3(_b6753)) →RΩ(1)
mark'(and'(_gen_true':mark':false':0':nil':ok'3(+(1, _$n5501)), _gen_true':mark':false':0':nil':ok'3(_b6753))) →IH
mark'(_*4)

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Lemmas:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)
first'(_gen_true':mark':false':0':nil':ok'3(+(1, _n2463)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n2463)
and'(_gen_true':mark':false':0':nil':ok'3(+(1, _n5500)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5500)

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

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

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

Proved the following rewrite lemma:
if'(_gen_true':mark':false':0':nil':ok'3(+(1, _n8660)), _gen_true':mark':false':0':nil':ok'3(b), _gen_true':mark':false':0':nil':ok'3(c)) → _*4, rt ∈ Ω(n8660)

Induction Base:
if'(_gen_true':mark':false':0':nil':ok'3(+(1, 0)), _gen_true':mark':false':0':nil':ok'3(b), _gen_true':mark':false':0':nil':ok'3(c))

Induction Step:
if'(_gen_true':mark':false':0':nil':ok'3(+(1, +(_$n8661, 1))), _gen_true':mark':false':0':nil':ok'3(_b11131), _gen_true':mark':false':0':nil':ok'3(_c11132)) →RΩ(1)
mark'(if'(_gen_true':mark':false':0':nil':ok'3(+(1, _$n8661)), _gen_true':mark':false':0':nil':ok'3(_b11131), _gen_true':mark':false':0':nil':ok'3(_c11132))) →IH
mark'(_*4)

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Lemmas:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)
first'(_gen_true':mark':false':0':nil':ok'3(+(1, _n2463)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n2463)
and'(_gen_true':mark':false':0':nil':ok'3(+(1, _n5500)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5500)
if'(_gen_true':mark':false':0':nil':ok'3(+(1, _n8660)), _gen_true':mark':false':0':nil':ok'3(b), _gen_true':mark':false':0':nil':ok'3(c)) → _*4, rt ∈ Ω(n8660)

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

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

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

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Lemmas:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)
first'(_gen_true':mark':false':0':nil':ok'3(+(1, _n2463)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n2463)
and'(_gen_true':mark':false':0':nil':ok'3(+(1, _n5500)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5500)
if'(_gen_true':mark':false':0':nil':ok'3(+(1, _n8660)), _gen_true':mark':false':0':nil':ok'3(b), _gen_true':mark':false':0':nil':ok'3(c)) → _*4, rt ∈ Ω(n8660)

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

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

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

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Lemmas:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)
first'(_gen_true':mark':false':0':nil':ok'3(+(1, _n2463)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n2463)
and'(_gen_true':mark':false':0':nil':ok'3(+(1, _n5500)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5500)
if'(_gen_true':mark':false':0':nil':ok'3(+(1, _n8660)), _gen_true':mark':false':0':nil':ok'3(b), _gen_true':mark':false':0':nil':ok'3(c)) → _*4, rt ∈ Ω(n8660)

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

The following defined symbols remain to be analysed:
top'

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

Rules:
active'(and'(true', X)) → mark'(X)
active'(and'(false', Y)) → mark'(false')
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
and' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
true' :: true':mark':false':0':nil':ok'
mark' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
false' :: true':mark':false':0':nil':ok'
if' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
add' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
0' :: true':mark':false':0':nil':ok'
s' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
first' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
nil' :: true':mark':false':0':nil':ok'
cons' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
from' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
proper' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
ok' :: true':mark':false':0':nil':ok' → true':mark':false':0':nil':ok'
top' :: true':mark':false':0':nil':ok' → top'
_hole_true':mark':false':0':nil':ok'1 :: true':mark':false':0':nil':ok'
_hole_top'2 :: top'
_gen_true':mark':false':0':nil':ok'3 :: Nat → true':mark':false':0':nil':ok'

Lemmas:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)
first'(_gen_true':mark':false':0':nil':ok'3(+(1, _n2463)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n2463)
and'(_gen_true':mark':false':0':nil':ok'3(+(1, _n5500)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n5500)
if'(_gen_true':mark':false':0':nil':ok'3(+(1, _n8660)), _gen_true':mark':false':0':nil':ok'3(b), _gen_true':mark':false':0':nil':ok'3(c)) → _*4, rt ∈ Ω(n8660)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
add'(_gen_true':mark':false':0':nil':ok'3(+(1, _n11)), _gen_true':mark':false':0':nil':ok'3(b)) → _*4, rt ∈ Ω(n11)