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(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__b → b

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
mark(g(X)) →⁺ a__g(mark(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / g(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(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__b → b

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__b → b

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c:f

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

Heuristically decided to analyse the following defined symbols:
a__f, mark

They will be analysed ascendingly in the following order:
a__f < mark

(8) Obligation:

TRS:
Rules:
a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__b → b

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c:f

Generator Equations:
gen_g:b:c:f2_0(0) ⇔ b
gen_g:b:c:f2_0(+(x, 1)) ⇔ g(gen_g:b:c:f2_0(x))

The following defined symbols remain to be analysed:
a__f, mark

They will be analysed ascendingly in the following order:
a__f < mark

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

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

(10) Obligation:

TRS:
Rules:
a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__b → b

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c:f

Generator Equations:
gen_g:b:c:f2_0(0) ⇔ b
gen_g:b:c:f2_0(+(x, 1)) ⇔ g(gen_g:b:c:f2_0(x))

The following defined symbols remain to be analysed:
mark

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

Proved the following rewrite lemma:
mark(gen_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n101₀)

Induction Base:
mark(gen_g:b:c:f2_0(+(1, 0)))

Induction Step:
mark(gen_g:b:c:f2_0(+(1, +(n101_0, 1)))) →_R^Ω(1)
a__g(mark(gen_g:b:c:f2_0(+(1, n101_0)))) →_IH
a__g(*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(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__b → b

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c:f

Lemmas:
mark(gen_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n101₀)

Generator Equations:
gen_g:b:c:f2_0(0) ⇔ b
gen_g:b:c:f2_0(+(x, 1)) ⇔ g(gen_g:b:c:f2_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_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n101₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__b → b

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c:f

Lemmas:
mark(gen_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n101₀)

Generator Equations:
gen_g:b:c:f2_0(0) ⇔ b
gen_g:b:c:f2_0(+(x, 1)) ⇔ g(gen_g:b:c:f2_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_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n101₀)

(18) BOUNDS(n^1, INF)