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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1

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

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

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

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

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:none
K tuples:

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

Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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