The Runtime Complexity (innermost) 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(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
top, check, f, match, proper, active

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

(6) Obligation:

Innermost TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found

Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
f(gen_mark:c:X:ok:found3_0(+(1, 0)))

Induction Step:
f(gen_mark:c:X:ok:found3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
mark(f(gen_mark:c:X:ok:found3_0(+(1, n5_0)))) →_IH
mark(*4_0)

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found

Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))

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

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

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found

Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))

The following defined symbols remain to be analysed:
match, top, check, active

They will be analysed ascendingly in the following order:
check < top
active < top
match < check

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

Innermost TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found

Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))

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

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

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

Innermost TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found

Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))

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

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

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Innermost TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found

Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))

The following defined symbols remain to be analysed:
top

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

Innermost TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found

Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))

Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found

Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(24) BOUNDS(n^1, INF)