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

active(f(X)) → mark(if(X, c, f(true)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(f(X)) → f(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(if(X1, X2, X3)) → if(X1, active(X2), X3)
f(mark(X)) → mark(f(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(X1, mark(X2), X3) → mark(if(X1, X2, X3))
proper(f(X)) → f(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
f(ok(X)) → ok(f(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
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'(f'(X)) → mark'(if'(X, c', f'(true')))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(f'(X)) → f'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(if'(X1, X2, X3)) → if'(X1, active'(X2), X3)
f'(mark'(X)) → mark'(f'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
if'(X1, mark'(X2), X3) → mark'(if'(X1, X2, X3))
proper'(f'(X)) → f'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(c') → ok'(c')
proper'(true') → ok'(true')
proper'(false') → ok'(false')
f'(ok'(X)) → ok'(f'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(f'(X)) → mark'(if'(X, c', f'(true')))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(f'(X)) → f'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(if'(X1, X2, X3)) → if'(X1, active'(X2), X3)
f'(mark'(X)) → mark'(f'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
if'(X1, mark'(X2), X3) → mark'(if'(X1, X2, X3))
proper'(f'(X)) → f'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(c') → ok'(c')
proper'(true') → ok'(true')
proper'(false') → ok'(false')
f'(ok'(X)) → ok'(f'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: c':true':mark':false':ok' → c':true':mark':false':ok'
f' :: c':true':mark':false':ok' → c':true':mark':false':ok'
mark' :: c':true':mark':false':ok' → c':true':mark':false':ok'
if' :: c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok'
c' :: c':true':mark':false':ok'
true' :: c':true':mark':false':ok'
false' :: c':true':mark':false':ok'
proper' :: c':true':mark':false':ok' → c':true':mark':false':ok'
ok' :: c':true':mark':false':ok' → c':true':mark':false':ok'
top' :: c':true':mark':false':ok' → top'
_hole_c':true':mark':false':ok'1 :: c':true':mark':false':ok'
_hole_top'2 :: top'
_gen_c':true':mark':false':ok'3 :: Nat → c':true':mark':false':ok'

Heuristically decided to analyse the following defined symbols:
active', if', f', proper', top'

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

Rules:
active'(f'(X)) → mark'(if'(X, c', f'(true')))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(f'(X)) → f'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(if'(X1, X2, X3)) → if'(X1, active'(X2), X3)
f'(mark'(X)) → mark'(f'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
if'(X1, mark'(X2), X3) → mark'(if'(X1, X2, X3))
proper'(f'(X)) → f'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(c') → ok'(c')
proper'(true') → ok'(true')
proper'(false') → ok'(false')
f'(ok'(X)) → ok'(f'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: c':true':mark':false':ok' → c':true':mark':false':ok'
f' :: c':true':mark':false':ok' → c':true':mark':false':ok'
mark' :: c':true':mark':false':ok' → c':true':mark':false':ok'
if' :: c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok'
c' :: c':true':mark':false':ok'
true' :: c':true':mark':false':ok'
false' :: c':true':mark':false':ok'
proper' :: c':true':mark':false':ok' → c':true':mark':false':ok'
ok' :: c':true':mark':false':ok' → c':true':mark':false':ok'
top' :: c':true':mark':false':ok' → top'
_hole_c':true':mark':false':ok'1 :: c':true':mark':false':ok'
_hole_top'2 :: top'
_gen_c':true':mark':false':ok'3 :: Nat → c':true':mark':false':ok'

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

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

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

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

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

Induction Step:
if'(_gen_c':true':mark':false':ok'3(+(1, +(_\$n6, 1))), _gen_c':true':mark':false':ok'3(_b1342), _gen_c':true':mark':false':ok'3(_c1343)) →RΩ(1)
mark'(if'(_gen_c':true':mark':false':ok'3(+(1, _\$n6)), _gen_c':true':mark':false':ok'3(_b1342), _gen_c':true':mark':false':ok'3(_c1343))) →IH
mark'(_*4)

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

Rules:
active'(f'(X)) → mark'(if'(X, c', f'(true')))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(f'(X)) → f'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(if'(X1, X2, X3)) → if'(X1, active'(X2), X3)
f'(mark'(X)) → mark'(f'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
if'(X1, mark'(X2), X3) → mark'(if'(X1, X2, X3))
proper'(f'(X)) → f'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(c') → ok'(c')
proper'(true') → ok'(true')
proper'(false') → ok'(false')
f'(ok'(X)) → ok'(f'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: c':true':mark':false':ok' → c':true':mark':false':ok'
f' :: c':true':mark':false':ok' → c':true':mark':false':ok'
mark' :: c':true':mark':false':ok' → c':true':mark':false':ok'
if' :: c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok'
c' :: c':true':mark':false':ok'
true' :: c':true':mark':false':ok'
false' :: c':true':mark':false':ok'
proper' :: c':true':mark':false':ok' → c':true':mark':false':ok'
ok' :: c':true':mark':false':ok' → c':true':mark':false':ok'
top' :: c':true':mark':false':ok' → top'
_hole_c':true':mark':false':ok'1 :: c':true':mark':false':ok'
_hole_top'2 :: top'
_gen_c':true':mark':false':ok'3 :: Nat → c':true':mark':false':ok'

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

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

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

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

Proved the following rewrite lemma:
f'(_gen_c':true':mark':false':ok'3(+(1, _n2469))) → _*4, rt ∈ Ω(n2469)

Induction Base:
f'(_gen_c':true':mark':false':ok'3(+(1, 0)))

Induction Step:
f'(_gen_c':true':mark':false':ok'3(+(1, +(_\$n2470, 1)))) →RΩ(1)
mark'(f'(_gen_c':true':mark':false':ok'3(+(1, _\$n2470)))) →IH
mark'(_*4)

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

Rules:
active'(f'(X)) → mark'(if'(X, c', f'(true')))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(f'(X)) → f'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(if'(X1, X2, X3)) → if'(X1, active'(X2), X3)
f'(mark'(X)) → mark'(f'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
if'(X1, mark'(X2), X3) → mark'(if'(X1, X2, X3))
proper'(f'(X)) → f'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(c') → ok'(c')
proper'(true') → ok'(true')
proper'(false') → ok'(false')
f'(ok'(X)) → ok'(f'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: c':true':mark':false':ok' → c':true':mark':false':ok'
f' :: c':true':mark':false':ok' → c':true':mark':false':ok'
mark' :: c':true':mark':false':ok' → c':true':mark':false':ok'
if' :: c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok'
c' :: c':true':mark':false':ok'
true' :: c':true':mark':false':ok'
false' :: c':true':mark':false':ok'
proper' :: c':true':mark':false':ok' → c':true':mark':false':ok'
ok' :: c':true':mark':false':ok' → c':true':mark':false':ok'
top' :: c':true':mark':false':ok' → top'
_hole_c':true':mark':false':ok'1 :: c':true':mark':false':ok'
_hole_top'2 :: top'
_gen_c':true':mark':false':ok'3 :: Nat → c':true':mark':false':ok'

Lemmas:
if'(_gen_c':true':mark':false':ok'3(+(1, _n5)), _gen_c':true':mark':false':ok'3(b), _gen_c':true':mark':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
f'(_gen_c':true':mark':false':ok'3(+(1, _n2469))) → _*4, rt ∈ Ω(n2469)

Generator Equations:
_gen_c':true':mark':false':ok'3(0) ⇔ c'
_gen_c':true':mark':false':ok'3(+(x, 1)) ⇔ mark'(_gen_c':true':mark':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'(f'(X)) → mark'(if'(X, c', f'(true')))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(f'(X)) → f'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(if'(X1, X2, X3)) → if'(X1, active'(X2), X3)
f'(mark'(X)) → mark'(f'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
if'(X1, mark'(X2), X3) → mark'(if'(X1, X2, X3))
proper'(f'(X)) → f'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(c') → ok'(c')
proper'(true') → ok'(true')
proper'(false') → ok'(false')
f'(ok'(X)) → ok'(f'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: c':true':mark':false':ok' → c':true':mark':false':ok'
f' :: c':true':mark':false':ok' → c':true':mark':false':ok'
mark' :: c':true':mark':false':ok' → c':true':mark':false':ok'
if' :: c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok'
c' :: c':true':mark':false':ok'
true' :: c':true':mark':false':ok'
false' :: c':true':mark':false':ok'
proper' :: c':true':mark':false':ok' → c':true':mark':false':ok'
ok' :: c':true':mark':false':ok' → c':true':mark':false':ok'
top' :: c':true':mark':false':ok' → top'
_hole_c':true':mark':false':ok'1 :: c':true':mark':false':ok'
_hole_top'2 :: top'
_gen_c':true':mark':false':ok'3 :: Nat → c':true':mark':false':ok'

Lemmas:
if'(_gen_c':true':mark':false':ok'3(+(1, _n5)), _gen_c':true':mark':false':ok'3(b), _gen_c':true':mark':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
f'(_gen_c':true':mark':false':ok'3(+(1, _n2469))) → _*4, rt ∈ Ω(n2469)

Generator Equations:
_gen_c':true':mark':false':ok'3(0) ⇔ c'
_gen_c':true':mark':false':ok'3(+(x, 1)) ⇔ mark'(_gen_c':true':mark':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'(f'(X)) → mark'(if'(X, c', f'(true')))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(f'(X)) → f'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(if'(X1, X2, X3)) → if'(X1, active'(X2), X3)
f'(mark'(X)) → mark'(f'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
if'(X1, mark'(X2), X3) → mark'(if'(X1, X2, X3))
proper'(f'(X)) → f'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(c') → ok'(c')
proper'(true') → ok'(true')
proper'(false') → ok'(false')
f'(ok'(X)) → ok'(f'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: c':true':mark':false':ok' → c':true':mark':false':ok'
f' :: c':true':mark':false':ok' → c':true':mark':false':ok'
mark' :: c':true':mark':false':ok' → c':true':mark':false':ok'
if' :: c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok'
c' :: c':true':mark':false':ok'
true' :: c':true':mark':false':ok'
false' :: c':true':mark':false':ok'
proper' :: c':true':mark':false':ok' → c':true':mark':false':ok'
ok' :: c':true':mark':false':ok' → c':true':mark':false':ok'
top' :: c':true':mark':false':ok' → top'
_hole_c':true':mark':false':ok'1 :: c':true':mark':false':ok'
_hole_top'2 :: top'
_gen_c':true':mark':false':ok'3 :: Nat → c':true':mark':false':ok'

Lemmas:
if'(_gen_c':true':mark':false':ok'3(+(1, _n5)), _gen_c':true':mark':false':ok'3(b), _gen_c':true':mark':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
f'(_gen_c':true':mark':false':ok'3(+(1, _n2469))) → _*4, rt ∈ Ω(n2469)

Generator Equations:
_gen_c':true':mark':false':ok'3(0) ⇔ c'
_gen_c':true':mark':false':ok'3(+(x, 1)) ⇔ mark'(_gen_c':true':mark':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'(f'(X)) → mark'(if'(X, c', f'(true')))
active'(if'(true', X, Y)) → mark'(X)
active'(if'(false', X, Y)) → mark'(Y)
active'(f'(X)) → f'(active'(X))
active'(if'(X1, X2, X3)) → if'(active'(X1), X2, X3)
active'(if'(X1, X2, X3)) → if'(X1, active'(X2), X3)
f'(mark'(X)) → mark'(f'(X))
if'(mark'(X1), X2, X3) → mark'(if'(X1, X2, X3))
if'(X1, mark'(X2), X3) → mark'(if'(X1, X2, X3))
proper'(f'(X)) → f'(proper'(X))
proper'(if'(X1, X2, X3)) → if'(proper'(X1), proper'(X2), proper'(X3))
proper'(c') → ok'(c')
proper'(true') → ok'(true')
proper'(false') → ok'(false')
f'(ok'(X)) → ok'(f'(X))
if'(ok'(X1), ok'(X2), ok'(X3)) → ok'(if'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: c':true':mark':false':ok' → c':true':mark':false':ok'
f' :: c':true':mark':false':ok' → c':true':mark':false':ok'
mark' :: c':true':mark':false':ok' → c':true':mark':false':ok'
if' :: c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok' → c':true':mark':false':ok'
c' :: c':true':mark':false':ok'
true' :: c':true':mark':false':ok'
false' :: c':true':mark':false':ok'
proper' :: c':true':mark':false':ok' → c':true':mark':false':ok'
ok' :: c':true':mark':false':ok' → c':true':mark':false':ok'
top' :: c':true':mark':false':ok' → top'
_hole_c':true':mark':false':ok'1 :: c':true':mark':false':ok'
_hole_top'2 :: top'
_gen_c':true':mark':false':ok'3 :: Nat → c':true':mark':false':ok'

Lemmas:
if'(_gen_c':true':mark':false':ok'3(+(1, _n5)), _gen_c':true':mark':false':ok'3(b), _gen_c':true':mark':false':ok'3(c)) → _*4, rt ∈ Ω(n5)
f'(_gen_c':true':mark':false':ok'3(+(1, _n2469))) → _*4, rt ∈ Ω(n2469)

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

No more defined symbols left to analyse.

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