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:

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

b, f, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

b, f, c

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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