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(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, x))))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(f, a(g, a(f, z0))) → a(f, a(g, a(g, a(f, z0))))
a(g, a(f, a(g, z0))) → a(g, a(f, a(f, a(g, z0))))

Tuples:

A(f, a(g, a(f, z0))) → c(A(f, a(g, a(g, a(f, z0)))), A(g, a(g, a(f, z0))), A(g, a(f, z0)), A(f, z0))
A(g, a(f, a(g, z0))) → c1(A(g, a(f, a(f, a(g, z0)))), A(f, a(f, a(g, z0))), A(f, a(g, z0)), A(g, z0))

S tuples:

A(f, a(g, a(f, z0))) → c(A(f, a(g, a(g, a(f, z0)))), A(g, a(g, a(f, z0))), A(g, a(f, z0)), A(f, z0))
A(g, a(f, a(g, z0))) → c1(A(g, a(f, a(f, a(g, z0)))), A(f, a(f, a(g, z0))), A(f, a(g, z0)), A(g, z0))

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1

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

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

A(f, a(g, a(f, x0))) → c(A(g, a(f, x0)), A(f, x0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(f, a(g, a(f, z0))) → a(f, a(g, a(g, a(f, z0))))
a(g, a(f, a(g, z0))) → a(g, a(f, a(f, a(g, z0))))

Tuples:

A(g, a(f, a(g, z0))) → c1(A(g, a(f, a(f, a(g, z0)))), A(f, a(f, a(g, z0))), A(f, a(g, z0)), A(g, z0))
A(f, a(g, a(f, x0))) → c(A(g, a(f, x0)), A(f, x0))

S tuples:

A(g, a(f, a(g, z0))) → c1(A(g, a(f, a(f, a(g, z0)))), A(f, a(f, a(g, z0))), A(f, a(g, z0)), A(g, z0))
A(f, a(g, a(f, x0))) → c(A(g, a(f, x0)), A(f, x0))

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1, c

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

Use narrowing to replace A(g, a(f, a(g, z0))) → c1(A(g, a(f, a(f, a(g, z0)))), A(f, a(f, a(g, z0))), A(f, a(g, z0)), A(g, z0)) by

A(g, a(f, a(g, x0))) → c1(A(f, a(g, x0)), A(g, x0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(f, a(g, a(f, z0))) → a(f, a(g, a(g, a(f, z0))))
a(g, a(f, a(g, z0))) → a(g, a(f, a(f, a(g, z0))))

Tuples:

A(f, a(g, a(f, x0))) → c(A(g, a(f, x0)), A(f, x0))
A(g, a(f, a(g, x0))) → c1(A(f, a(g, x0)), A(g, x0))

S tuples:

A(f, a(g, a(f, x0))) → c(A(g, a(f, x0)), A(f, x0))
A(g, a(f, a(g, x0))) → c1(A(f, a(g, x0)), A(g, x0))

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1

(7) 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(f, a(g, a(f, x0))) → c(A(g, a(f, x0)), A(f, x0))
A(g, a(f, a(g, x0))) → c1(A(f, a(g, x0)), A(g, x0))

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

A(f, a(g, a(f, x0))) → c(A(g, a(f, x0)), A(f, x0))
A(g, a(f, a(g, x0))) → c1(A(f, a(g, x0)), A(g, x0))

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

POL(A(x₁, x₂)) = [1] + x₂   
POL(a(x₁, x₂)) = [2] + [4]x₂   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(f) = 0   
POL(g) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(f, a(g, a(f, z0))) → a(f, a(g, a(g, a(f, z0))))
a(g, a(f, a(g, z0))) → a(g, a(f, a(f, a(g, z0))))

Tuples:

A(f, a(g, a(f, x0))) → c(A(g, a(f, x0)), A(f, x0))
A(g, a(f, a(g, x0))) → c1(A(f, a(g, x0)), A(g, x0))

S tuples:none
K tuples:

A(f, a(g, a(f, x0))) → c(A(g, a(f, x0)), A(f, x0))
A(g, a(f, a(g, x0))) → c1(A(f, a(g, x0)), A(g, x0))

Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))