### (0) Obligation:

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

active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(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(X1), X2, X3) →+ mark(f(X1, X2, X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].

### (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(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

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

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

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

### (8) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Generator Equations:
gen_a:b:mark:c:ok3_0(0) ⇔ a
gen_a:b:mark:c:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:c: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

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

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

Induction Base:
f(gen_a:b:mark:c:ok3_0(+(1, 0)), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(c))

Induction Step:
f(gen_a:b:mark:c:ok3_0(+(1, +(n5_0, 1))), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(c)) →RΩ(1)
mark(f(gen_a:b:mark:c:ok3_0(+(1, n5_0)), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(c))) →IH
mark(*4_0)

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

### (11) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(+(1, n5_0)), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(c)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_a:b:mark:c:ok3_0(0) ⇔ a
gen_a:b:mark:c:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:c: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

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

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

### (13) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(+(1, n5_0)), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(c)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_a:b:mark:c:ok3_0(0) ⇔ a
gen_a:b:mark:c:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:c:ok3_0(x))

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

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

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

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

### (15) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(+(1, n5_0)), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(c)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_a:b:mark:c:ok3_0(0) ⇔ a
gen_a:b:mark:c:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:c:ok3_0(x))

The following defined symbols remain to be analysed:
top

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

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

### (17) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(+(1, n5_0)), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(c)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_a:b:mark:c:ok3_0(0) ⇔ a
gen_a:b:mark:c:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:c:ok3_0(x))

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

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

### (20) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(active(X1), X2, X3)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(mark(X1), X2, X3) → mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(+(1, n5_0)), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(c)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_a:b:mark:c:ok3_0(0) ⇔ a
gen_a:b:mark:c:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:c:ok3_0(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

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