The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

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

a(b(a(b(x)))) → b(a(b(a(a(b(x))))))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(a(b(z0)))) → b(a(b(a(a(b(z0))))))

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c

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

Use narrowing to replace A(b(a(b(z0)))) → c(A(b(a(a(b(z0))))), A(a(b(z0))), A(b(z0))) by

A(b(a(b(x0)))) → c(A(b(x0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(a(b(z0)))) → b(a(b(a(a(b(z0))))))

Tuples:

A(b(a(b(x0)))) → c(A(b(x0)))

S tuples:

A(b(a(b(x0)))) → c(A(b(x0)))

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

A(b(a(b(x0)))) → c(A(b(x0)))

We considered the (Usable) Rules:none
And the Tuples:

A(b(a(b(x0)))) → c(A(b(x0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A(x₁)) = [2]x₁   
POL(a(x₁)) = [1] + [4]x₁   
POL(b(x₁)) = x₁   
POL(c(x₁)) = x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(a(b(z0)))) → b(a(b(a(a(b(z0))))))

Tuples:

A(b(a(b(x0)))) → c(A(b(x0)))

S tuples:none
K tuples:

A(b(a(b(x0)))) → c(A(b(x0)))

Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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