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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

f, c

Defined Pair Symbols:

F

Compound Symbols:

c1, c2

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

f, c

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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