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:

add(0, x) → x
add(s(x), y) → s(add(x, y))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))

Tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))

S tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))

K tuples:none
Defined Rule Symbols:

add

Defined Pair Symbols:

ADD

Compound Symbols:

c1

(3) 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.

ADD(s(z0), z1) → c1(ADD(z0, z1))

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

ADD(s(z0), z1) → c1(ADD(z0, z1))

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

POL(ADD(x₁, x₂)) = [5]x₁   
POL(c1(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))

Tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))

S tuples:none
K tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))

Defined Rule Symbols:

add

Defined Pair Symbols:

ADD

Compound Symbols:

c1

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))