(0) Obligation:

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

a(x1) → x1
a(x1) → b(c(x1))
a(b(b(x1))) → b(b(a(a(x1))))
b(x1) → 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(c(z0))
a(b(b(z0))) → b(b(a(a(z0))))
b(z0) → z0
Tuples:

A(z0) → c2(B(c(z0)))
A(b(b(z0))) → c3(B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
S tuples:

A(z0) → c2(B(c(z0)))
A(b(b(z0))) → c3(B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

A

Compound Symbols:

c2, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

A(b(b(z0))) → c3(B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(z0) → c2(B(c(z0)))
S tuples:

A(z0) → c2(B(c(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(c(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(z0) → b(c(z0))
a(b(b(z0))) → b(b(a(a(z0))))
b(z0) → 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))