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:

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A → c1(G(c))
F(g(z0), b) → c3(F(a, z0), A)

S tuples:

A → c1(G(c))
F(g(z0), b) → c3(F(a, z0), A)

K tuples:none
Defined Rule Symbols:

a, g, f

Defined Pair Symbols:

A, F

Compound Symbols:

c1, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(g(z0), b) → c3(F(a, z0), A)

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A → c1(G(c))

S tuples:

A → c1(G(c))

K tuples:none
Defined Rule Symbols:

a, g, f

Defined Pair Symbols:

A

Compound Symbols:

c1

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

Removed 1 trailing nodes:

A → c1(G(c))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

a, g, f

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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