(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

*(x, +(y, z)) → +(*(x, y), *(x, z))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
Tuples:

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
S tuples:

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:

*'

Compound Symbols:

c

(3) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
S tuples:

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

*'

Compound Symbols:

c

(5) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(*'(x1, x2)) = [3] + [4]x2
POL(+(x1, x2)) = [4] + x1 + x2
POL(c(x1, x2)) = x1 + x2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
S tuples:none
K tuples:

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

*'

Compound Symbols:

c

(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty