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(b(a, z)) → z
b(y, b(a, z)) → b(f(c(y, y, a)), b(f(z), a))
f(f(f(c(z, x, a)))) → b(f(x), z)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(f(f(c(z0, z1, a)))) → c2(B(f(z1), z0), F(z1))
B(z0, b(a, z1)) → c3(B(f(c(z0, z0, a)), b(f(z1), a)), F(c(z0, z0, a)), B(f(z1), a), F(z1))

S tuples:

F(f(f(c(z0, z1, a)))) → c2(B(f(z1), z0), F(z1))
B(z0, b(a, z1)) → c3(B(f(c(z0, z0, a)), b(f(z1), a)), F(c(z0, z0, a)), B(f(z1), a), F(z1))

K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:

F, B

Compound Symbols:

c2, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(f(c(z0, z1, a)))) → c2(B(f(z1), z0), F(z1))
B(z0, b(a, z1)) → c3(B(f(c(z0, z0, a)), b(f(z1), a)), F(c(z0, z0, a)), B(f(z1), a), F(z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

f, b

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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