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(b, X, c) → a__f(X, a__c, X)
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(c) → a__c
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(b, X2, c)) →⁺ a__f(mark(X2), a__c, mark(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X2 / f(b, X2, c)].
The result substitution is [ ].

The rewrite sequence
mark(f(b, X2, c)) →⁺ a__f(mark(X2), a__c, mark(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [2].
The pumping substitution is [X2 / f(b, X2, c)].
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(b, X, c) → a__f(X, a__c, X)
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(c) → a__c
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(b, X, c) → a__f(X, a__c, X)
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(c) → a__c
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c

Types:
a__f :: b:c:f → b:c:f → b:c:f → b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f → b:c:f
f :: b:c:f → b:c:f → b:c:f → b:c:f
hole_b:c:f1_0 :: b:c:f
gen_b:c:f2_0 :: Nat → 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(b, X, c) → a__f(X, a__c, X)
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(c) → a__c
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c

Types:
a__f :: b:c:f → b:c:f → b:c:f → b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f → b:c:f
f :: b:c:f → b:c:f → b:c:f → b:c:f
hole_b:c:f1_0 :: b:c:f
gen_b:c:f2_0 :: Nat → b:c:f

Generator Equations:
gen_b:c:f2_0(0) ⇔ b
gen_b:c:f2_0(+(x, 1)) ⇔ f(b, gen_b:c:f2_0(x), b)

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

Types:
a__f :: b:c:f → b:c:f → b:c:f → b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f → b:c:f
f :: b:c:f → b:c:f → b:c:f → b:c:f
hole_b:c:f1_0 :: b:c:f
gen_b:c:f2_0 :: Nat → b:c:f

Generator Equations:
gen_b:c:f2_0(0) ⇔ b
gen_b:c:f2_0(+(x, 1)) ⇔ f(b, gen_b:c:f2_0(x), b)

The following defined symbols remain to be analysed:
mark

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

Proved the following rewrite lemma:
mark(gen_b:c:f2_0(n16_0)) → gen_b:c:f2_0(n16_0), rt ∈ Ω(1 + n16₀)

Induction Base:
mark(gen_b:c:f2_0(0)) →_R^Ω(1)
b

Induction Step:
mark(gen_b:c:f2_0(+(n16_0, 1))) →_R^Ω(1)
a__f(b, mark(gen_b:c:f2_0(n16_0)), b) →_IH
a__f(b, gen_b:c:f2_0(c17_0), b) →_R^Ω(1)
f(b, gen_b:c:f2_0(n16_0), b)

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

Types:
a__f :: b:c:f → b:c:f → b:c:f → b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f → b:c:f
f :: b:c:f → b:c:f → b:c:f → b:c:f
hole_b:c:f1_0 :: b:c:f
gen_b:c:f2_0 :: Nat → b:c:f

Lemmas:
mark(gen_b:c:f2_0(n16_0)) → gen_b:c:f2_0(n16_0), rt ∈ Ω(1 + n16₀)

Generator Equations:
gen_b:c:f2_0(0) ⇔ b
gen_b:c:f2_0(+(x, 1)) ⇔ f(b, gen_b:c:f2_0(x), b)

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_b:c:f2_0(n16_0)) → gen_b:c:f2_0(n16_0), rt ∈ Ω(1 + n16₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
a__f :: b:c:f → b:c:f → b:c:f → b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f → b:c:f
f :: b:c:f → b:c:f → b:c:f → b:c:f
hole_b:c:f1_0 :: b:c:f
gen_b:c:f2_0 :: Nat → b:c:f

Lemmas:
mark(gen_b:c:f2_0(n16_0)) → gen_b:c:f2_0(n16_0), rt ∈ Ω(1 + n16₀)

Generator Equations:
gen_b:c:f2_0(0) ⇔ b
gen_b:c:f2_0(+(x, 1)) ⇔ f(b, gen_b:c:f2_0(x), b)

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_b:c:f2_0(n16_0)) → gen_b:c:f2_0(n16_0), rt ∈ Ω(1 + n16₀)

(18) BOUNDS(n^1, INF)