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, i(x)) → 0
+(+(x, y), z) → +(x, +(y, z))
*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(x, y), z) → +(*(x, z), *(y, z))

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, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))

Tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

S tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

K tuples:none
Defined Rule Symbols:

+, *

Defined Pair Symbols:

+', *'

Compound Symbols:

c2, c3, c4

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(z0, 0) → z0
+(z0, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))

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

+, *

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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