(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) CdtUnreachableProof (EQUIVALENT transformation)

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

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))

(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:

B(z0, z1) → c1(A(c(z1), a(0, z0)), A(0, z0))
S tuples:

B(z0, z1) → c1(A(c(z1), a(0, z0)), A(0, z0))
K tuples:none
Defined Rule Symbols:

b, a

Defined Pair Symbols:

B

Compound Symbols:

c1

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

Removed 1 trailing nodes:

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

(6) 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

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))