(0) Obligation:

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

f(f(a)) → f(g(n__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 CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → f(g(n__f(a)))
f(z0) → n__f(z0)
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:

F(f(a)) → c(F(g(n__f(a))))
ACTIVATE(n__f(z0)) → c2(F(z0))
S tuples:

F(f(a)) → c(F(g(n__f(a))))
ACTIVATE(n__f(z0)) → c2(F(z0))
K tuples:none
Defined Rule Symbols:

f, activate

Defined Pair Symbols:

F, ACTIVATE

Compound Symbols:

c, c2

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

F(f(a)) → c(F(g(n__f(a))))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → f(g(n__f(a)))
f(z0) → n__f(z0)
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c2(F(z0))
S tuples:

ACTIVATE(n__f(z0)) → c2(F(z0))
K tuples:none
Defined Rule Symbols:

f, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c2

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

ACTIVATE(n__f(z0)) → c2(F(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → f(g(n__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

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))