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:

g(X) → u(h(X), h(X), X)
u(d, c(Y), X) → k(Y)
h(d) → c(a)
h(d) → c(b)
f(k(a), k(b), X) → f(X, X, X)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0) → u(h(z0), h(z0), z0)
u(d, c(z0), z1) → k(z0)
h(d) → c(a)
h(d) → c(b)
f(k(a), k(b), z0) → f(z0, z0, z0)

Tuples:

G(z0) → c1(U(h(z0), h(z0), z0), H(z0), H(z0))
F(k(a), k(b), z0) → c5(F(z0, z0, z0))

S tuples:

G(z0) → c1(U(h(z0), h(z0), z0), H(z0), H(z0))
F(k(a), k(b), z0) → c5(F(z0, z0, z0))

K tuples:none
Defined Rule Symbols:

g, u, h, f

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c5

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

Removed 2 trailing nodes:

F(k(a), k(b), z0) → c5(F(z0, z0, z0))
G(z0) → c1(U(h(z0), h(z0), z0), H(z0), H(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0) → u(h(z0), h(z0), z0)
u(d, c(z0), z1) → k(z0)
h(d) → c(a)
h(d) → c(b)
f(k(a), k(b), z0) → f(z0, z0, z0)

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

g, u, h, f

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))