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:

a__f(f(a)) → c(f(g(f(a))))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c(X)) → c(X)
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

a__f(f(a)) → c(f(g(f(a))))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c(X)) → c(X)
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
c/0

(6) Obligation:

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

a__f(f(a)) → c
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c) → c
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
a__f(f(a)) → c
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c) → c
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

Types:
a__f :: a:f:c:g → a:f:c:g
f :: a:f:c:g → a:f:c:g
a :: a:f:c:g
c :: a:f:c:g
mark :: a:f:c:g → a:f:c:g
g :: a:f:c:g → a:f:c:g
hole_a:f:c:g1_0 :: a:f:c:g
gen_a:f:c:g2_0 :: Nat → a:f:c:g

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
mark

(10) Obligation:

TRS:
Rules:
a__f(f(a)) → c
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c) → c
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

Types:
a__f :: a:f:c:g → a:f:c:g
f :: a:f:c:g → a:f:c:g
a :: a:f:c:g
c :: a:f:c:g
mark :: a:f:c:g → a:f:c:g
g :: a:f:c:g → a:f:c:g
hole_a:f:c:g1_0 :: a:f:c:g
gen_a:f:c:g2_0 :: Nat → a:f:c:g

Generator Equations:
gen_a:f:c:g2_0(0) ⇔ a
gen_a:f:c:g2_0(+(x, 1)) ⇔ f(gen_a:f:c:g2_0(x))

The following defined symbols remain to be analysed:
mark

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_a:f:c:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
mark(gen_a:f:c:g2_0(+(1, 0)))

Induction Step:
mark(gen_a:f:c:g2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
a__f(mark(gen_a:f:c:g2_0(+(1, n4_0)))) →_IH
a__f(*3_0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
a__f(f(a)) → c
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c) → c
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

Types:
a__f :: a:f:c:g → a:f:c:g
f :: a:f:c:g → a:f:c:g
a :: a:f:c:g
c :: a:f:c:g
mark :: a:f:c:g → a:f:c:g
g :: a:f:c:g → a:f:c:g
hole_a:f:c:g1_0 :: a:f:c:g
gen_a:f:c:g2_0 :: Nat → a:f:c:g

Lemmas:
mark(gen_a:f:c:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_a:f:c:g2_0(0) ⇔ a
gen_a:f:c:g2_0(+(x, 1)) ⇔ f(gen_a:f:c:g2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_a:f:c:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
a__f(f(a)) → c
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c) → c
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

Types:
a__f :: a:f:c:g → a:f:c:g
f :: a:f:c:g → a:f:c:g
a :: a:f:c:g
c :: a:f:c:g
mark :: a:f:c:g → a:f:c:g
g :: a:f:c:g → a:f:c:g
hole_a:f:c:g1_0 :: a:f:c:g
gen_a:f:c:g2_0 :: Nat → a:f:c:g

Lemmas:
mark(gen_a:f:c:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_a:f:c:g2_0(0) ⇔ a
gen_a:f:c:g2_0(+(x, 1)) ⇔ f(gen_a:f:c:g2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_a:f:c:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(18) BOUNDS(n^1, INF)