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, y), *(a, y)) → *(+(x, a), y)
*(*(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:

+(*(z0, z1), *(a, z1)) → *(+(z0, a), z1)
*(*(z0, z1), z2) → *(z0, *(z1, z2))

Tuples:

+'(*(z0, z1), *(a, z1)) → c(*'(+(z0, a), z1), +'(z0, a))
*'(*(z0, z1), z2) → c1(*'(z0, *(z1, z2)), *'(z1, z2))

S tuples:

+'(*(z0, z1), *(a, z1)) → c(*'(+(z0, a), z1), +'(z0, a))
*'(*(z0, z1), z2) → c1(*'(z0, *(z1, z2)), *'(z1, z2))

K tuples:none
Defined Rule Symbols:

+, *

Defined Pair Symbols:

+', *'

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

+'(*(z0, z1), *(a, z1)) → c(*'(+(z0, a), z1), +'(z0, a))
*'(*(z0, z1), z2) → c1(*'(z0, *(z1, z2)), *'(z1, z2))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(a, z1)) → *(+(z0, a), z1)
*(*(z0, z1), z2) → *(z0, *(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))