The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(mark(X)) →⁺ mark(f(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative 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))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
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:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok

(7) OrderProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

TRS:
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:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok

Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(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

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
if(gen_c:true:mark:false:ok3_0(+(1, 0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c))

Induction Step:
if(gen_c:true:mark:false:ok3_0(+(1, +(n5_0, 1))), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) →_R^Ω(1)
mark(if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
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:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok

Lemmas:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_c:true:mark:false:ok3_0(+(1, n1591_0))) → *4_0, rt ∈ Ω(n1591₀)

Induction Base:
f(gen_c:true:mark:false:ok3_0(+(1, 0)))

Induction Step:
f(gen_c:true:mark:false:ok3_0(+(1, +(n1591_0, 1)))) →_R^Ω(1)
mark(f(gen_c:true:mark:false:ok3_0(+(1, n1591_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
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:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok

Lemmas:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
f(gen_c:true:mark:false:ok3_0(+(1, n1591_0))) → *4_0, rt ∈ Ω(n1591₀)

Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(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

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
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:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok

Lemmas:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
f(gen_c:true:mark:false:ok3_0(+(1, n1591_0))) → *4_0, rt ∈ Ω(n1591₀)

Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(x))

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

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

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
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:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok

Lemmas:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
f(gen_c:true:mark:false:ok3_0(+(1, n1591_0))) → *4_0, rt ∈ Ω(n1591₀)

Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(x))

The following defined symbols remain to be analysed:
top

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

TRS:
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:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok

Lemmas:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
f(gen_c:true:mark:false:ok3_0(+(1, n1591_0))) → *4_0, rt ∈ Ω(n1591₀)

Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

TRS:
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:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok

Lemmas:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
f(gen_c:true:mark:false:ok3_0(+(1, n1591_0))) → *4_0, rt ∈ Ω(n1591₀)

Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

TRS:
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:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok

Lemmas:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
if(gen_c:true:mark:false:ok3_0(+(1, n5_0)), gen_c:true:mark:false:ok3_0(b), gen_c:true:mark:false:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)

(28) BOUNDS(n^1, INF)