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:

and(tt, X) → X
plus(N, 0) → N
plus(N, s(M)) → s(plus(N, M))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(tt, z0) → z0
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))

Tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

S tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

K tuples:none
Defined Rule Symbols:

and, plus

Defined Pair Symbols:

PLUS

Compound Symbols:

c2

(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.

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

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

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

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

POL(PLUS(x₁, x₂)) = [5]x₂   
POL(c2(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(tt, z0) → z0
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))

Tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

S tuples:none
K tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

Defined Rule Symbols:

and, plus

Defined Pair Symbols:

PLUS

Compound Symbols:

c2

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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