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(x, y) → c(a(c(y), a(0, x)))
a(y, x) → y
a(y, c(b(a(0, x), 0))) → b(a(c(b(0, y)), x), 0)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → c(a(c(z1), a(0, z0)))
a(z0, z1) → z0
a(z0, c(b(a(0, z1), 0))) → b(a(c(b(0, z0)), z1), 0)

Tuples:

B(z0, z1) → c1(A(c(z1), a(0, z0)), A(0, z0))
A(z0, c(b(a(0, z1), 0))) → c3(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0))

S tuples:

B(z0, z1) → c1(A(c(z1), a(0, z0)), A(0, z0))
A(z0, c(b(a(0, z1), 0))) → c3(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0))

K tuples:none
Defined Rule Symbols:

b, a

Defined Pair Symbols:

B, A

Compound Symbols:

c1, c3

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

A(z0, c(b(a(0, z1), 0))) → c3(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0))
B(z0, z1) → c1(A(c(z1), a(0, z0)), A(0, z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → c(a(c(z1), a(0, z0)))
a(z0, z1) → z0
a(z0, c(b(a(0, z1), 0))) → b(a(c(b(0, z0)), z1), 0)

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

b, a

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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