(0) Obligation:

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

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

F(f(z0)) → c1(C(n__f(g(n__f(z0)))))
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))
S tuples:

F(f(z0)) → c1(C(n__f(g(n__f(z0)))))
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))
K tuples:none
Defined Rule Symbols:

f, c, h, d, activate

Defined Pair Symbols:

F, C, H, ACTIVATE

Compound Symbols:

c1, c3, c4, c6, c7

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(z0)) → c1(C(n__f(g(n__f(z0)))))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))
S tuples:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))
K tuples:none
Defined Rule Symbols:

f, c, h, d, activate

Defined Pair Symbols:

C, H, ACTIVATE

Compound Symbols:

c3, c4, c6, c7

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 4 of 4 dangling nodes:

C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(n__f(g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
d(z0) → n__d(z0)
activate(n__f(z0)) → f(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, c, h, d, activate

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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