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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtUnreachableProof (⇔)
↳4 CdtProblem
↳5 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳6 CdtProblem
↳7 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳10 CdtProblem
↳11 SIsEmptyProof (⇔)
↳12 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

F(f(a)) → c(F(g(n__f(n__a))))
ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__a) → c4(A)
S tuples:

F(f(a)) → c(F(g(n__f(n__a))))
ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__a) → c4(A)
K tuples:none
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

F, ACTIVATE

Compound Symbols:

c, c3, c4

(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(n__a))))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__a) → c4(A)
S tuples:

ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__a) → c4(A)
K tuples:none
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 2 dangling nodes:

ACTIVATE(n__a) → c4(A)

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3

(7) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [5]x1   
POL(c3(x1)) = x1   
POL(n__f(x1)) = [1] + x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0))
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3

(11) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(12) BOUNDS(O(1), O(1))