The Runtime Complexity (innermost) 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__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

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

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

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

(8) Obligation:

Innermost TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

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

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

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

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

(10) Obligation:

Innermost TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

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

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

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

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

(12) Obligation:

Innermost TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

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

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

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

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

(14) Obligation:

Innermost TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

The following defined symbols remain to be analysed:
mark

They will be analysed ascendingly in the following order:
a__h = a__g
a__h = mark
a__h = a__f
a__g = mark
a__g = a__f
mark = a__f

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

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

(16) Obligation:

Innermost TRS:
Rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Types:
a__h :: a:b:h:g:f → a:b:h:g:f
a__g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
mark :: a:b:h:g:f → a:b:h:g:f
a :: a:b:h:g:f
a__f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
b :: a:b:h:g:f
a__a :: a:b:h:g:f
h :: a:b:h:g:f → a:b:h:g:f
g :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
f :: a:b:h:g:f → a:b:h:g:f → a:b:h:g:f
hole_a:b:h:g:f1_0 :: a:b:h:g:f
gen_a:b:h:g:f2_0 :: Nat → a:b:h:g:f

Generator Equations:
gen_a:b:h:g:f2_0(0) ⇔ a
gen_a:b:h:g:f2_0(+(x, 1)) ⇔ h(gen_a:b:h:g:f2_0(x))

No more defined symbols left to analyse.