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:

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(a, f(a, z0)) → c(F(a, f(f(a, z0), f(a, a))), F(f(a, z0), f(a, a)), F(a, z0), F(a, a))

S tuples:

F(a, f(a, z0)) → c(F(a, f(f(a, z0), f(a, a))), F(f(a, z0), f(a, a)), F(a, z0), F(a, a))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

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

F(a, f(a, x0)) → c(F(a, x0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(a, f(a, x0)) → c(F(a, x0))

S tuples:

F(a, f(a, x0)) → c(F(a, x0))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

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.

F(a, f(a, x0)) → c(F(a, x0))

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

F(a, f(a, x0)) → c(F(a, x0))

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

POL(F(x₁, x₂)) = [2]x₂   
POL(a) = [3]   
POL(c(x₁)) = x₁   
POL(f(x₁, x₂)) = [1] + [4]x₂   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(a, f(a, x0)) → c(F(a, x0))

S tuples:none
K tuples:

F(a, f(a, x0)) → c(F(a, x0))

Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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