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), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳8 CdtProblem
↳9 CdtUsableRulesProof (⇔, 0 ms)
↳10 CdtProblem
↳11 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 43 ms)
↳12 CdtProblem
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 BOUNDS(O(1), O(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 Cpx (relative) TRS 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)))))
F(z0) → c2
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
G(z0) → c5
D(z0) → c6
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c8(G(z0))
ACTIVATE(n__d(z0)) → c9(D(z0))
ACTIVATE(z0) → c10
S tuples:

F(f(z0)) → c1(C(n__f(n__g(n__f(z0)))))
F(z0) → c2
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
G(z0) → c5
D(z0) → c6
ACTIVATE(n__f(z0)) → c7(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c8(G(z0))
ACTIVATE(n__d(z0)) → c9(D(z0))
ACTIVATE(z0) → c10
K tuples:none
Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

F, C, H, G, D, ACTIVATE

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10

(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 6 trailing nodes:

ACTIVATE(z0) → c10
G(z0) → c5
F(z0) → c2
ACTIVATE(n__d(z0)) → c9(D(z0))
D(z0) → c6
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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(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(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
S tuples:

C(z0) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

C, ACTIVATE

Compound Symbols:

c3, c7

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

C(z0) → c3(ACTIVATE(z0))

(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:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, c, h, g, d, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7

(11) CdtRuleRemovalProof (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(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [5]x1   
POL(c7(x1)) = x1   
POL(n__f(x1)) = [1] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__f(z0)) → c7(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7

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

The set S is empty

(14) BOUNDS(O(1), O(1))