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:

app(nil, xs) → nil
app(cons(x, xs), ys) → cons(x, app(xs, ys))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → nil
app(cons(z0, z1), z2) → cons(z0, app(z1, z2))

Tuples:

APP(cons(z0, z1), z2) → c1(APP(z1, z2))

S tuples:

APP(cons(z0, z1), z2) → c1(APP(z1, z2))

K tuples:none
Defined Rule Symbols:

app

Defined Pair Symbols:

APP

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.

APP(cons(z0, z1), z2) → c1(APP(z1, z2))

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

APP(cons(z0, z1), z2) → c1(APP(z1, z2))

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

POL(APP(x₁, x₂)) = [5]x₁   
POL(c1(x₁)) = x₁   
POL(cons(x₁, x₂)) = [1] + x₂   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → nil
app(cons(z0, z1), z2) → cons(z0, app(z1, z2))

Tuples:

APP(cons(z0, z1), z2) → c1(APP(z1, z2))

S tuples:none
K tuples:

APP(cons(z0, z1), z2) → c1(APP(z1, z2))

Defined Rule Symbols:

app

Defined Pair Symbols:

APP

Compound Symbols:

c1

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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