### (0) Obligation:

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

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

Types:
active :: c:mark:ok → c:mark:ok
c :: c:mark:ok
mark :: c:mark:ok → c:mark:ok
f :: c:mark:ok → c:mark:ok
g :: c:mark:ok → c:mark:ok
proper :: c:mark:ok → c:mark:ok
ok :: c:mark:ok → c:mark:ok
top :: c:mark:ok → top
hole_c:mark:ok1_0 :: c:mark:ok
hole_top2_0 :: top
gen_c:mark:ok3_0 :: Nat → c:mark:ok

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

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

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

### (8) Obligation:

TRS:
Rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: c:mark:ok → c:mark:ok
c :: c:mark:ok
mark :: c:mark:ok → c:mark:ok
f :: c:mark:ok → c:mark:ok
g :: c:mark:ok → c:mark:ok
proper :: c:mark:ok → c:mark:ok
ok :: c:mark:ok → c:mark:ok
top :: c:mark:ok → top
hole_c:mark:ok1_0 :: c:mark:ok
hole_top2_0 :: top
gen_c:mark:ok3_0 :: Nat → c:mark:ok

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

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

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

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

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

### (10) Obligation:

TRS:
Rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: c:mark:ok → c:mark:ok
c :: c:mark:ok
mark :: c:mark:ok → c:mark:ok
f :: c:mark:ok → c:mark:ok
g :: c:mark:ok → c:mark:ok
proper :: c:mark:ok → c:mark:ok
ok :: c:mark:ok → c:mark:ok
top :: c:mark:ok → top
hole_c:mark:ok1_0 :: c:mark:ok
hole_top2_0 :: top
gen_c:mark:ok3_0 :: Nat → c:mark:ok

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

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

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

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

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

### (12) Obligation:

TRS:
Rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: c:mark:ok → c:mark:ok
c :: c:mark:ok
mark :: c:mark:ok → c:mark:ok
f :: c:mark:ok → c:mark:ok
g :: c:mark:ok → c:mark:ok
proper :: c:mark:ok → c:mark:ok
ok :: c:mark:ok → c:mark:ok
top :: c:mark:ok → top
hole_c:mark:ok1_0 :: c:mark:ok
hole_top2_0 :: top
gen_c:mark:ok3_0 :: Nat → c:mark:ok

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

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

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

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

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

### (14) Obligation:

TRS:
Rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: c:mark:ok → c:mark:ok
c :: c:mark:ok
mark :: c:mark:ok → c:mark:ok
f :: c:mark:ok → c:mark:ok
g :: c:mark:ok → c:mark:ok
proper :: c:mark:ok → c:mark:ok
ok :: c:mark:ok → c:mark:ok
top :: c:mark:ok → top
hole_c:mark:ok1_0 :: c:mark:ok
hole_top2_0 :: top
gen_c:mark:ok3_0 :: Nat → c:mark:ok

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

The following defined symbols remain to be analysed:
top

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

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

### (16) Obligation:

TRS:
Rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: c:mark:ok → c:mark:ok
c :: c:mark:ok
mark :: c:mark:ok → c:mark:ok
f :: c:mark:ok → c:mark:ok
g :: c:mark:ok → c:mark:ok
proper :: c:mark:ok → c:mark:ok
ok :: c:mark:ok → c:mark:ok
top :: c:mark:ok → top
hole_c:mark:ok1_0 :: c:mark:ok
hole_top2_0 :: top
gen_c:mark:ok3_0 :: Nat → c:mark:ok

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

No more defined symbols left to analyse.