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:

f(X, g(X), Y) → f(Y, Y, Y)
g(b) → c
b → c

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, g(z0), z1) → f(z1, z1, z1)
g(b) → c
b → c

Tuples:

F(z0, g(z0), z1) → c1(F(z1, z1, z1))

S tuples:

F(z0, g(z0), z1) → c1(F(z1, z1, z1))

K tuples:none
Defined Rule Symbols:

f, g, b

Defined Pair Symbols:

F

Compound Symbols:

c1

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

Removed 1 trailing nodes:

F(z0, g(z0), z1) → c1(F(z1, z1, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, g(z0), z1) → f(z1, z1, z1)
g(b) → c
b → c

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

f, g, b

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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