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, z)) → +(+(x, y), z)
*(x, +(y, z)) → +(*(x, y), *(x, z))
+(+(x, *(y, z)), *(y, u)) → +(x, *(y, +(z, u)))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(+(z0, *(z1, z2)), *(z1, z3)) → +(z0, *(z1, +(z2, z3)))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))

Tuples:

+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(+(z0, *(z1, z2)), *(z1, z3)) → c1(+'(z0, *(z1, +(z2, z3))), *'(z1, +(z2, z3)), +'(z2, z3))
*'(z0, +(z1, z2)) → c2(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))

S tuples:

+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(+(z0, *(z1, z2)), *(z1, z3)) → c1(+'(z0, *(z1, +(z2, z3))), *'(z1, +(z2, z3)), +'(z2, z3))
*'(z0, +(z1, z2)) → c2(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))

K tuples:none
Defined Rule Symbols:

+, *

Defined Pair Symbols:

+', *'

Compound Symbols:

c, c1, c2

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(+(z0, *(z1, z2)), *(z1, z3)) → c1(+'(z0, *(z1, +(z2, z3))), *'(z1, +(z2, z3)), +'(z2, z3))
*'(z0, +(z1, z2)) → c2(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(+(z0, *(z1, z2)), *(z1, z3)) → +(z0, *(z1, +(z2, z3)))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, 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))