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

(0) Obligation:

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

f(f(a)) → c(n__f(g(f(a))))
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → c(n__f(g(f(a))))
f(z0) → n__f(z0)
activate(n__f(z0)) → f(z0)
activate(z0) → z0

Tuples:

F(f(a)) → c1(F(a))
F(z0) → c2
ACTIVATE(n__f(z0)) → c3(F(z0))
ACTIVATE(z0) → c4

S tuples:

F(f(a)) → c1(F(a))
F(z0) → c2
ACTIVATE(n__f(z0)) → c3(F(z0))
ACTIVATE(z0) → c4

K tuples:none
Defined Rule Symbols:

f, activate

Defined Pair Symbols:

F, ACTIVATE

Compound Symbols:

c1, c2, c3, c4

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

Removed 4 trailing nodes:

F(f(a)) → c1(F(a))
F(z0) → c2
ACTIVATE(z0) → c4
ACTIVATE(n__f(z0)) → c3(F(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → c(n__f(g(f(a))))
f(z0) → n__f(z0)
activate(n__f(z0)) → f(z0)
activate(z0) → z0

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

f, activate

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(6) BOUNDS(1, 1)