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__f(a, b, X) → a__f(X, X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

a__f(a, b, X) → a__f(X, X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
a__f(a, b, X) → a__f(X, X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c

Types:
a__f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c → a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
c :: a:b:f:c
hole_a:b:f:c1_0 :: a:b:f:c
gen_a:b:f:c2_0 :: Nat → a:b:f:c

(5) 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

(6) Obligation:

Innermost TRS:
Rules:
a__f(a, b, X) → a__f(X, X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c

Types:
a__f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c → a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
c :: a:b:f:c
hole_a:b:f:c1_0 :: a:b:f:c
gen_a:b:f:c2_0 :: Nat → a:b:f:c

Generator Equations:
gen_a:b:f:c2_0(0) ⇔ a
gen_a:b:f:c2_0(+(x, 1)) ⇔ f(a, a, gen_a:b:f:c2_0(x))

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_a:b:f:c2_0(n4_0)) → gen_a:b:f:c2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
mark(gen_a:b:f:c2_0(0)) →_R^Ω(1)
a

Induction Step:
mark(gen_a:b:f:c2_0(+(n4_0, 1))) →_R^Ω(1)
a__f(a, a, mark(gen_a:b:f:c2_0(n4_0))) →_IH
a__f(a, a, gen_a:b:f:c2_0(c5_0)) →_R^Ω(1)
f(a, a, gen_a:b:f:c2_0(n4_0))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
a__f(a, b, X) → a__f(X, X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c

Types:
a__f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c → a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
c :: a:b:f:c
hole_a:b:f:c1_0 :: a:b:f:c
gen_a:b:f:c2_0 :: Nat → a:b:f:c

Lemmas:
mark(gen_a:b:f:c2_0(n4_0)) → gen_a:b:f:c2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_a:b:f:c2_0(0) ⇔ a
gen_a:b:f:c2_0(+(x, 1)) ⇔ f(a, a, gen_a:b:f:c2_0(x))

The following defined symbols remain to be analysed:
a__f

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

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
a__f(a, b, X) → a__f(X, X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c

Types:
a__f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c → a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
c :: a:b:f:c
hole_a:b:f:c1_0 :: a:b:f:c
gen_a:b:f:c2_0 :: Nat → a:b:f:c

Lemmas:
mark(gen_a:b:f:c2_0(n4_0)) → gen_a:b:f:c2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_a:b:f:c2_0(0) ⇔ a
gen_a:b:f:c2_0(+(x, 1)) ⇔ f(a, a, gen_a:b:f:c2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_a:b:f:c2_0(n4_0)) → gen_a:b:f:c2_0(n4_0), rt ∈ Ω(1 + n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
a__f(a, b, X) → a__f(X, X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c

Types:
a__f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c → a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c → a:b:f:c → a:b:f:c → a:b:f:c
c :: a:b:f:c
hole_a:b:f:c1_0 :: a:b:f:c
gen_a:b:f:c2_0 :: Nat → a:b:f:c

Lemmas:
mark(gen_a:b:f:c2_0(n4_0)) → gen_a:b:f:c2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_a:b:f:c2_0(0) ⇔ a
gen_a:b:f:c2_0(+(x, 1)) ⇔ f(a, a, gen_a:b:f:c2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_a:b:f:c2_0(n4_0)) → gen_a:b:f:c2_0(n4_0), rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)