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

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(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'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Heuristically decided to analyse the following defined symbols:
active', if', zero', s', prod', fact', p', add', proper', top'

They will be analysed ascendingly in the following order:
if' < active'
zero' < active'
s' < active'
prod' < active'
fact' < active'
p' < active'
active' < top'
if' < proper'
zero' < proper'
s' < proper'
prod' < proper'
fact' < proper'
p' < proper'
proper' < top'

Rules:
active'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))

The following defined symbols remain to be analysed:
if', active', zero', s', prod', fact', p', add', proper', top'

They will be analysed ascendingly in the following order:
if' < active'
zero' < active'
s' < active'
prod' < active'
fact' < active'
p' < active'
active' < top'
if' < proper'
zero' < proper'
s' < proper'
prod' < proper'
fact' < proper'
p' < proper'
proper' < top'

Proved the following rewrite lemma:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)

Induction Base:
if'(_gen_0':mark':true':false':ok'3(+(1, 0)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c))

Induction Step:
if'(_gen_0':mark':true':false':ok'3(+(1, +(_\$n6, 1))), _gen_0':mark':true':false':ok'3(_b1018), _gen_0':mark':true':false':ok'3(_c1019)) →RΩ(1)
mark'(if'(_gen_0':mark':true':false':ok'3(+(1, _\$n6)), _gen_0':mark':true':false':ok'3(_b1018), _gen_0':mark':true':false':ok'3(_c1019))) →IH
mark'(_*4)

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

Rules:
active'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))

The following defined symbols remain to be analysed:
zero', active', s', prod', fact', p', add', proper', top'

They will be analysed ascendingly in the following order:
zero' < active'
s' < active'
prod' < active'
fact' < active'
p' < active'
active' < top'
zero' < proper'
s' < proper'
prod' < proper'
fact' < proper'
p' < proper'
proper' < top'

Proved the following rewrite lemma:
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)

Induction Base:
zero'(_gen_0':mark':true':false':ok'3(+(1, 0)))

Induction Step:
zero'(_gen_0':mark':true':false':ok'3(+(1, +(_\$n3691, 1)))) →RΩ(1)
mark'(zero'(_gen_0':mark':true':false':ok'3(+(1, _\$n3691)))) →IH
mark'(_*4)

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

Rules:
active'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))

The following defined symbols remain to be analysed:
s', active', prod', fact', p', add', proper', top'

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

Proved the following rewrite lemma:
s'(_gen_0':mark':true':false':ok'3(+(1, _n5543))) → _*4, rt ∈ Ω(n5543)

Induction Base:
s'(_gen_0':mark':true':false':ok'3(+(1, 0)))

Induction Step:
s'(_gen_0':mark':true':false':ok'3(+(1, +(_\$n5544, 1)))) →RΩ(1)
mark'(s'(_gen_0':mark':true':false':ok'3(+(1, _\$n5544)))) →IH
mark'(_*4)

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

Rules:
active'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)
s'(_gen_0':mark':true':false':ok'3(+(1, _n5543))) → _*4, rt ∈ Ω(n5543)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))

The following defined symbols remain to be analysed:
prod', active', fact', p', add', proper', top'

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

Proved the following rewrite lemma:
prod'(_gen_0':mark':true':false':ok'3(+(1, _n7520)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n7520)

Induction Base:
prod'(_gen_0':mark':true':false':ok'3(+(1, 0)), _gen_0':mark':true':false':ok'3(b))

Induction Step:
prod'(_gen_0':mark':true':false':ok'3(+(1, +(_\$n7521, 1))), _gen_0':mark':true':false':ok'3(_b9205)) →RΩ(1)
mark'(prod'(_gen_0':mark':true':false':ok'3(+(1, _\$n7521)), _gen_0':mark':true':false':ok'3(_b9205))) →IH
mark'(_*4)

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

Rules:
active'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)
s'(_gen_0':mark':true':false':ok'3(+(1, _n5543))) → _*4, rt ∈ Ω(n5543)
prod'(_gen_0':mark':true':false':ok'3(+(1, _n7520)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n7520)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))

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

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

Proved the following rewrite lemma:
fact'(_gen_0':mark':true':false':ok'3(+(1, _n11299))) → _*4, rt ∈ Ω(n11299)

Induction Base:
fact'(_gen_0':mark':true':false':ok'3(+(1, 0)))

Induction Step:
fact'(_gen_0':mark':true':false':ok'3(+(1, +(_\$n11300, 1)))) →RΩ(1)
mark'(fact'(_gen_0':mark':true':false':ok'3(+(1, _\$n11300)))) →IH
mark'(_*4)

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

Rules:
active'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)
s'(_gen_0':mark':true':false':ok'3(+(1, _n5543))) → _*4, rt ∈ Ω(n5543)
prod'(_gen_0':mark':true':false':ok'3(+(1, _n7520)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n7520)
fact'(_gen_0':mark':true':false':ok'3(+(1, _n11299))) → _*4, rt ∈ Ω(n11299)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))

The following defined symbols remain to be analysed:

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

Proved the following rewrite lemma:
p'(_gen_0':mark':true':false':ok'3(+(1, _n13591))) → _*4, rt ∈ Ω(n13591)

Induction Base:
p'(_gen_0':mark':true':false':ok'3(+(1, 0)))

Induction Step:
p'(_gen_0':mark':true':false':ok'3(+(1, +(_\$n13592, 1)))) →RΩ(1)
mark'(p'(_gen_0':mark':true':false':ok'3(+(1, _\$n13592)))) →IH
mark'(_*4)

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

Rules:
active'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)
s'(_gen_0':mark':true':false':ok'3(+(1, _n5543))) → _*4, rt ∈ Ω(n5543)
prod'(_gen_0':mark':true':false':ok'3(+(1, _n7520)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n7520)
fact'(_gen_0':mark':true':false':ok'3(+(1, _n11299))) → _*4, rt ∈ Ω(n11299)
p'(_gen_0':mark':true':false':ok'3(+(1, _n13591))) → _*4, rt ∈ Ω(n13591)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))

The following defined symbols remain to be analysed:

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

Proved the following rewrite lemma:
add'(_gen_0':mark':true':false':ok'3(+(1, _n16007)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n16007)

Induction Base:

Induction Step:
mark'(_*4)

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

Rules:
active'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)
s'(_gen_0':mark':true':false':ok'3(+(1, _n5543))) → _*4, rt ∈ Ω(n5543)
prod'(_gen_0':mark':true':false':ok'3(+(1, _n7520)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n7520)
fact'(_gen_0':mark':true':false':ok'3(+(1, _n11299))) → _*4, rt ∈ Ω(n11299)
p'(_gen_0':mark':true':false':ok'3(+(1, _n13591))) → _*4, rt ∈ Ω(n13591)
add'(_gen_0':mark':true':false':ok'3(+(1, _n16007)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n16007)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':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'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)
s'(_gen_0':mark':true':false':ok'3(+(1, _n5543))) → _*4, rt ∈ Ω(n5543)
prod'(_gen_0':mark':true':false':ok'3(+(1, _n7520)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n7520)
fact'(_gen_0':mark':true':false':ok'3(+(1, _n11299))) → _*4, rt ∈ Ω(n11299)
p'(_gen_0':mark':true':false':ok'3(+(1, _n13591))) → _*4, rt ∈ Ω(n13591)
add'(_gen_0':mark':true':false':ok'3(+(1, _n16007)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n16007)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':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'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)
s'(_gen_0':mark':true':false':ok'3(+(1, _n5543))) → _*4, rt ∈ Ω(n5543)
prod'(_gen_0':mark':true':false':ok'3(+(1, _n7520)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n7520)
fact'(_gen_0':mark':true':false':ok'3(+(1, _n11299))) → _*4, rt ∈ Ω(n11299)
p'(_gen_0':mark':true':false':ok'3(+(1, _n13591))) → _*4, rt ∈ Ω(n13591)
add'(_gen_0':mark':true':false':ok'3(+(1, _n16007)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n16007)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':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'(fact'(X)) → mark'(if'(zero'(X), s'(0'), prod'(X, fact'(p'(X)))))
active'(prod'(0', X)) → mark'(0')
active'(prod'(s'(X), Y)) → mark'(add'(Y, prod'(X, Y)))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(zero'(0')) → mark'(true')
active'(zero'(s'(X))) → mark'(false')
active'(p'(s'(X))) → mark'(X)
active'(fact'(X)) → fact'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(zero'(X)) → zero'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(prod'(X1, X2)) → prod'(active'(X1), X2)
active'(prod'(X1, X2)) → prod'(X1, active'(X2))
active'(p'(X)) → p'(active'(X))
fact'(mark'(X)) → mark'(fact'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
zero'(mark'(X)) → mark'(zero'(X))
s'(mark'(X)) → mark'(s'(X))
prod'(mark'(X1), X2) → mark'(prod'(X1, X2))
prod'(X1, mark'(X2)) → mark'(prod'(X1, X2))
p'(mark'(X)) → mark'(p'(X))
proper'(fact'(X)) → fact'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(zero'(X)) → zero'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(prod'(X1, X2)) → prod'(proper'(X1), proper'(X2))
proper'(p'(X)) → p'(proper'(X))
proper'(true') → ok'(true')
proper'(false') → ok'(false')
fact'(ok'(X)) → ok'(fact'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
zero'(ok'(X)) → ok'(zero'(X))
s'(ok'(X)) → ok'(s'(X))
prod'(ok'(X1), ok'(X2)) → ok'(prod'(X1, X2))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
fact' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
mark' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
if' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
zero' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
s' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
0' :: 0':mark':true':false':ok'
prod' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
p' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
add' :: 0':mark':true':false':ok' → 0':mark':true':false':ok' → 0':mark':true':false':ok'
true' :: 0':mark':true':false':ok'
false' :: 0':mark':true':false':ok'
proper' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
ok' :: 0':mark':true':false':ok' → 0':mark':true':false':ok'
top' :: 0':mark':true':false':ok' → top'
_hole_0':mark':true':false':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'

Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
zero'(_gen_0':mark':true':false':ok'3(+(1, _n3690))) → _*4, rt ∈ Ω(n3690)
s'(_gen_0':mark':true':false':ok'3(+(1, _n5543))) → _*4, rt ∈ Ω(n5543)
prod'(_gen_0':mark':true':false':ok'3(+(1, _n7520)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n7520)
fact'(_gen_0':mark':true':false':ok'3(+(1, _n11299))) → _*4, rt ∈ Ω(n11299)
p'(_gen_0':mark':true':false':ok'3(+(1, _n13591))) → _*4, rt ∈ Ω(n13591)
add'(_gen_0':mark':true':false':ok'3(+(1, _n16007)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n16007)

Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
if'(_gen_0':mark':true':false':ok'3(+(1, _n5)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n5)