The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

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

f(X, X) → f(a, n__b)
b → a
b → n__b
activate(n__b) → b
activate(X) → X

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(a, n__b)
b → a
b → n__b
activate(n__b) → b
activate(z0) → z0

Tuples:

F(z0, z0) → c(F(a, n__b))
ACTIVATE(n__b) → c3(B)

S tuples:

F(z0, z0) → c(F(a, n__b))
ACTIVATE(n__b) → c3(B)

K tuples:none
Defined Rule Symbols:

f, b, activate

Defined Pair Symbols:

F, ACTIVATE

Compound Symbols:

c, c3

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

Removed 2 trailing nodes:

ACTIVATE(n__b) → c3(B)
F(z0, z0) → c(F(a, n__b))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(a, n__b)
b → a
b → n__b
activate(n__b) → b
activate(z0) → z0

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, b, activate

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))