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:

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(z0, z1, a), a, a) → b(z0, z1)
f(c(z0, z1, z2)) → c(z2, f(b(z1, z2)), a)
b(z0, b(c(a, z1, a), f(f(z2)))) → c(c(z1, a, z0), z0, z2)

Tuples:

C(c(z0, z1, a), a, a) → c1(B(z0, z1))
F(c(z0, z1, z2)) → c2(C(z2, f(b(z1, z2)), a), F(b(z1, z2)), B(z1, z2))
B(z0, b(c(a, z1, a), f(f(z2)))) → c3(C(c(z1, a, z0), z0, z2), C(z1, a, z0))

S tuples:

C(c(z0, z1, a), a, a) → c1(B(z0, z1))
F(c(z0, z1, z2)) → c2(C(z2, f(b(z1, z2)), a), F(b(z1, z2)), B(z1, z2))
B(z0, b(c(a, z1, a), f(f(z2)))) → c3(C(c(z1, a, z0), z0, z2), C(z1, a, z0))

K tuples:none
Defined Rule Symbols:

c, f, b

Defined Pair Symbols:

C, F, B

Compound Symbols:

c1, c2, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

C(c(z0, z1, a), a, a) → c1(B(z0, z1))
F(c(z0, z1, z2)) → c2(C(z2, f(b(z1, z2)), a), F(b(z1, z2)), B(z1, z2))
B(z0, b(c(a, z1, a), f(f(z2)))) → c3(C(c(z1, a, z0), z0, z2), C(z1, a, z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(z0, z1, a), a, a) → b(z0, z1)
f(c(z0, z1, z2)) → c(z2, f(b(z1, z2)), a)
b(z0, b(c(a, z1, a), f(f(z2)))) → c(c(z1, a, z0), z0, z2)

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

c, f, b

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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