The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

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

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

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
mark(f(a, b, X3)) →⁺ a__f(mark(mark(X3)), mark(X3), mark(mark(X3)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X3 / f(a, b, X3)].
The result substitution is [ ].

The rewrite sequence
mark(f(a, b, X3)) →⁺ a__f(mark(mark(X3)), mark(X3), mark(mark(X3)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [X3 / f(a, b, X3)].
The result substitution is [ ].

(2) BOUNDS(2^n, 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(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(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: FULL

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

Infered types.

(6) Obligation:

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

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

(9) 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(mark(a), a, mark(gen_a:b:f:c2_0(n4_0))) →_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).

(10) Complex Obligation (BEST)

(11) Obligation:

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

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

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

(14) 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)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

(17) 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)

(18) BOUNDS(n^1, INF)