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

(0) Obligation:

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

f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(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(z0)) → c(n__f(n__g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
g(z0) → n__g(z0)
d(z0) → n__d(z0)
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0

Tuples:

F(f(z0)) → c1(C(n__f(n__g(n__f(z0)))))
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c8(G(z0))
ACTIVATE(n__d(z0)) → c9(D(z0))

S tuples:

F(f(z0)) → c1(C(n__f(n__g(n__f(z0)))))
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c8(G(z0))
ACTIVATE(n__d(z0)) → c9(D(z0))

K tuples:none
Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

F, C, H, ACTIVATE

Compound Symbols:

c1, c3, c4, c7, c8, c9

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

F(f(z0)) → c1(C(n__f(n__g(n__f(z0)))))
H(z0) → c4(C(n__d(z0)))

Removed 2 trailing nodes:

ACTIVATE(n__d(z0)) → c9(D(z0))
ACTIVATE(n__g(z0)) → c8(G(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(n__f(n__g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
g(z0) → n__g(z0)
d(z0) → n__d(z0)
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0

Tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))

S tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

C, ACTIVATE

Compound Symbols:

c3, c7

(5) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(n__f(n__g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
g(z0) → n__g(z0)
d(z0) → n__d(z0)
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0

Tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))

S tuples:

ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))

K tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))

Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

C, ACTIVATE

Compound Symbols:

c3, c7

(7) 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)) → c7(F(activate(z0)), ACTIVATE(z0))

We considered the (Usable) Rules:

activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0
d(z0) → n__d(z0)
g(z0) → n__g(z0)
f(z0) → n__f(z0)

And the Tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x₁)) = x₁   
POL(C(x₁)) = [4] + [4]x₁   
POL(D(x₁)) = 0   
POL(F(x₁)) = 0   
POL(activate(x₁)) = [4]   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(d(x₁)) = [3] + [3]x₁   
POL(f(x₁)) = [3]   
POL(g(x₁)) = [3]x₁   
POL(n__d(x₁)) = [3]   
POL(n__f(x₁)) = [1] + x₁   
POL(n__g(x₁)) = [2]   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(n__f(n__g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
g(z0) → n__g(z0)
d(z0) → n__d(z0)
activate(n__f(z0)) → f(activate(z0))
activate(n__g(z0)) → g(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0

Tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))

S tuples:none
K tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))

Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

C, ACTIVATE

Compound Symbols:

c3, c7

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))