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

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 292 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 typed CpxTrs
    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 20 ms)
    ↳10 typed CpxTrs
    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
    ↳12 typed CpxTrs
    ↳13 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
    ↳14 typed CpxTrs
    ↳15 NoRewriteLemmaProof (LOWER BOUND(ID), 22 ms)
    ↳16 typed CpxTrs

(0) Obligation:

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

active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(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
g(ok(X)) →+ ok(g(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 [ ].

(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(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(X))
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(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:c:d:ok → mark:c:d:ok
g :: mark:c:d:ok → mark:c:d:ok
mark :: mark:c:d:ok → mark:c:d:ok
h :: mark:c:d:ok → mark:c:d:ok
c :: mark:c:d:ok
d :: mark:c:d:ok
proper :: mark:c:d:ok → mark:c:d:ok
ok :: mark:c:d:ok → mark:c:d:ok
top :: mark:c:d:ok → top
hole_mark:c:d:ok1_0 :: mark:c:d:ok
hole_top2_0 :: top
gen_mark:c:d:ok3_0 :: Nat → mark:c:d:ok

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

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

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

(8) Obligation:

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

Types:
active :: mark:c:d:ok → mark:c:d:ok
g :: mark:c:d:ok → mark:c:d:ok
mark :: mark:c:d:ok → mark:c:d:ok
h :: mark:c:d:ok → mark:c:d:ok
c :: mark:c:d:ok
d :: mark:c:d:ok
proper :: mark:c:d:ok → mark:c:d:ok
ok :: mark:c:d:ok → mark:c:d:ok
top :: mark:c:d:ok → top
hole_mark:c:d:ok1_0 :: mark:c:d:ok
hole_top2_0 :: top
gen_mark:c:d:ok3_0 :: Nat → mark:c:d:ok

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

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

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

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

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

(10) Obligation:

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

Types:
active :: mark:c:d:ok → mark:c:d:ok
g :: mark:c:d:ok → mark:c:d:ok
mark :: mark:c:d:ok → mark:c:d:ok
h :: mark:c:d:ok → mark:c:d:ok
c :: mark:c:d:ok
d :: mark:c:d:ok
proper :: mark:c:d:ok → mark:c:d:ok
ok :: mark:c:d:ok → mark:c:d:ok
top :: mark:c:d:ok → top
hole_mark:c:d:ok1_0 :: mark:c:d:ok
hole_top2_0 :: top
gen_mark:c:d:ok3_0 :: Nat → mark:c:d:ok

Generator Equations:
gen_mark:c:d:ok3_0(0) ⇔ c
gen_mark:c:d:ok3_0(+(x, 1)) ⇔ mark(gen_mark:c:d: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(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:c:d:ok → mark:c:d:ok
g :: mark:c:d:ok → mark:c:d:ok
mark :: mark:c:d:ok → mark:c:d:ok
h :: mark:c:d:ok → mark:c:d:ok
c :: mark:c:d:ok
d :: mark:c:d:ok
proper :: mark:c:d:ok → mark:c:d:ok
ok :: mark:c:d:ok → mark:c:d:ok
top :: mark:c:d:ok → top
hole_mark:c:d:ok1_0 :: mark:c:d:ok
hole_top2_0 :: top
gen_mark:c:d:ok3_0 :: Nat → mark:c:d:ok

Generator Equations:
gen_mark:c:d:ok3_0(0) ⇔ c
gen_mark:c:d:ok3_0(+(x, 1)) ⇔ mark(gen_mark:c:d: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(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:c:d:ok → mark:c:d:ok
g :: mark:c:d:ok → mark:c:d:ok
mark :: mark:c:d:ok → mark:c:d:ok
h :: mark:c:d:ok → mark:c:d:ok
c :: mark:c:d:ok
d :: mark:c:d:ok
proper :: mark:c:d:ok → mark:c:d:ok
ok :: mark:c:d:ok → mark:c:d:ok
top :: mark:c:d:ok → top
hole_mark:c:d:ok1_0 :: mark:c:d:ok
hole_top2_0 :: top
gen_mark:c:d:ok3_0 :: Nat → mark:c:d:ok

Generator Equations:
gen_mark:c:d:ok3_0(0) ⇔ c
gen_mark:c:d:ok3_0(+(x, 1)) ⇔ mark(gen_mark:c:d: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(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:c:d:ok → mark:c:d:ok
g :: mark:c:d:ok → mark:c:d:ok
mark :: mark:c:d:ok → mark:c:d:ok
h :: mark:c:d:ok → mark:c:d:ok
c :: mark:c:d:ok
d :: mark:c:d:ok
proper :: mark:c:d:ok → mark:c:d:ok
ok :: mark:c:d:ok → mark:c:d:ok
top :: mark:c:d:ok → top
hole_mark:c:d:ok1_0 :: mark:c:d:ok
hole_top2_0 :: top
gen_mark:c:d:ok3_0 :: Nat → mark:c:d:ok

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

No more defined symbols left to analyse.