(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a(x1) → x1
a(x1) → b(x1)
b(x1) → x1
b(a(c(x1))) → c(c(a(a(b(x1)))))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(z0) → b(z0)
b(z0) → z0
b(a(c(z0))) → c(c(a(a(b(z0)))))
Tuples:

A(z0) → c2(B(z0))
B(a(c(z0))) → c4(A(a(b(z0))), A(b(z0)), B(z0))
S tuples:

A(z0) → c2(B(z0))
B(a(c(z0))) → c4(A(a(b(z0))), A(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

A, B

Compound Symbols:

c2, c4

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

B(a(c(z0))) → c4(A(a(b(z0))), A(b(z0)), B(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(z0) → b(z0)
b(z0) → z0
b(a(c(z0))) → c(c(a(a(b(z0)))))
Tuples:

A(z0) → c2(B(z0))
S tuples:

A(z0) → c2(B(z0))
K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

A

Compound Symbols:

c2

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

A(z0) → c2(B(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(z0) → b(z0)
b(z0) → z0
b(a(c(z0))) → c(c(a(a(b(z0)))))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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