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:

U11(tt, M, N) → U12(tt, M, N)
U12(tt, M, N) → s(plus(N, M))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, z0, z1)
U12(tt, z0, z1) → s(plus(z1, z0))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)

Tuples:

U11'(tt, z0, z1) → c(U12'(tt, z0, z1))
U12'(tt, z0, z1) → c1(PLUS(z1, z0))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))

S tuples:

U11'(tt, z0, z1) → c(U12'(tt, z0, z1))
U12'(tt, z0, z1) → c1(PLUS(z1, z0))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))

K tuples:none
Defined Rule Symbols:

U11, U12, plus

Defined Pair Symbols:

U11', U12', PLUS

Compound Symbols:

c, c1, c3

(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.

U11'(tt, z0, z1) → c(U12'(tt, z0, z1))

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

U11'(tt, z0, z1) → c(U12'(tt, z0, z1))
U12'(tt, z0, z1) → c1(PLUS(z1, z0))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))

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

POL(PLUS(x₁, x₂)) = [2]x₂   
POL(U11'(x₁, x₂, x₃)) = [2] + [2]x₂   
POL(U12'(x₁, x₂, x₃)) = [4]x₁ + [2]x₂   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   
POL(tt) = 0   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, z0, z1)
U12(tt, z0, z1) → s(plus(z1, z0))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)

Tuples:

U11'(tt, z0, z1) → c(U12'(tt, z0, z1))
U12'(tt, z0, z1) → c1(PLUS(z1, z0))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))

S tuples:

U12'(tt, z0, z1) → c1(PLUS(z1, z0))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))

K tuples:

U11'(tt, z0, z1) → c(U12'(tt, z0, z1))

Defined Rule Symbols:

U11, U12, plus

Defined Pair Symbols:

U11', U12', PLUS

Compound Symbols:

c, c1, c3

(5) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

U12'(tt, z0, z1) → c1(PLUS(z1, z0))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, z0, z1))

Now S is empty

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