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) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) 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

(3) 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 [ ].

(4) BOUNDS(2^n, INF)