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:

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

+'(z0, s(z1)) → c1(S(+(z0, z1)), +'(z0, z1))
+'(0, s(z0)) → c2(S(z0))
S(+(0, z0)) → c3(S(z0))

S tuples:

+'(z0, s(z1)) → c1(S(+(z0, z1)), +'(z0, z1))
+'(0, s(z0)) → c2(S(z0))
S(+(0, z0)) → c3(S(z0))

K tuples:none
Defined Rule Symbols:

+, s

Defined Pair Symbols:

+', S

Compound Symbols:

c1, c2, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

+'(z0, s(z1)) → c1(S(+(z0, z1)), +'(z0, z1))
+'(0, s(z0)) → c2(S(z0))
S(+(0, z0)) → c3(S(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

+, s

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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