The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(+(x, 0)) → f(x)
+(x, +(y, z)) → +(+(x, y), z)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(+(z0, 0)) → f(z0)
+(z0, +(z1, z2)) → +(+(z0, z1), z2)

Tuples:

F(+(z0, 0)) → c(F(z0))
+'(z0, +(z1, z2)) → c1(+'(+(z0, z1), z2), +'(z0, z1))

S tuples:

F(+(z0, 0)) → c(F(z0))
+'(z0, +(z1, z2)) → c1(+'(+(z0, z1), z2), +'(z0, z1))

K tuples:none
Defined Rule Symbols:

f, +

Defined Pair Symbols:

F, +'

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

F(+(z0, 0)) → c(F(z0))
+'(z0, +(z1, z2)) → c1(+'(+(z0, z1), z2), +'(z0, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(+(z0, 0)) → f(z0)
+(z0, +(z1, z2)) → +(+(z0, z1), z2)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, +

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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