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(b) → f(b)
f(a) → g(a)
b → a

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(b) → f(b)
f(a) → g(a)
b → a

Tuples:

G(b) → c(F(b), B)
F(a) → c1(G(a))

S tuples:

G(b) → c(F(b), B)
F(a) → c1(G(a))

K tuples:none
Defined Rule Symbols:

g, f, b

Defined Pair Symbols:

G, F

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

G(b) → c(F(b), B)

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(b) → f(b)
f(a) → g(a)
b → a

Tuples:

F(a) → c1(G(a))

S tuples:

F(a) → c1(G(a))

K tuples:none
Defined Rule Symbols:

g, f, b

Defined Pair Symbols:

F

Compound Symbols:

c1

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

Removed 1 trailing nodes:

F(a) → c1(G(a))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(b) → f(b)
f(a) → g(a)
b → a

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

g, f, b

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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