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

(0) Obligation:

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

h(X) → g(X, X)
g(a, X) → f(b, activate(X))
f(X, X) → h(a)
a → b
activate(X) → X

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0) → g(z0, z0)
g(a, z0) → f(b, activate(z0))
f(z0, z0) → h(a)
a → b
activate(z0) → z0

Tuples:

H(z0) → c(G(z0, z0))
G(a, z0) → c1(F(b, activate(z0)), ACTIVATE(z0))
F(z0, z0) → c2(H(a), A)

S tuples:

H(z0) → c(G(z0, z0))
G(a, z0) → c1(F(b, activate(z0)), ACTIVATE(z0))
F(z0, z0) → c2(H(a), A)

K tuples:none
Defined Rule Symbols:

h, g, f, a, activate

Defined Pair Symbols:

H, G, F

Compound Symbols:

c, c1, c2

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

G(a, z0) → c1(F(b, activate(z0)), ACTIVATE(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0) → g(z0, z0)
g(a, z0) → f(b, activate(z0))
f(z0, z0) → h(a)
a → b
activate(z0) → z0

Tuples:

H(z0) → c(G(z0, z0))
F(z0, z0) → c2(H(a), A)

S tuples:

H(z0) → c(G(z0, z0))
F(z0, z0) → c2(H(a), A)

K tuples:none
Defined Rule Symbols:

h, g, f, a, activate

Defined Pair Symbols:

H, F

Compound Symbols:

c, c2

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

H(z0) → c(G(z0, z0))
F(z0, z0) → c2(H(a), A)

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0) → g(z0, z0)
g(a, z0) → f(b, activate(z0))
f(z0, z0) → h(a)
a → b
activate(z0) → z0

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

h, g, f, a, activate

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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