(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(a(b(f(z0))))
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:

F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
F(a(g(z0))) → c1(B(z0))
S tuples:

F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
F(a(g(z0))) → c1(B(z0))
K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:

F

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(a(b(f(z0))))
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:

F(a(g(z0))) → c1(B(z0))
S tuples:

F(a(g(z0))) → c1(B(z0))
K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:

F

Compound Symbols:

c1

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

F(a(g(z0))) → c1(B(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(a(b(f(z0))))
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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