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), +(x, z)) → *(x, +(y, z))
+(*(x, y), +(*(x, z), 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), +(z0, z2)) → *(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), z3)) → +(*(z0, +(z1, z2)), z3)

Tuples:

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

S tuples:

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))