The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

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

*(x, *(y, z)) → *(otimes(x, y), z)
*(1, y) → y
*(+(x, y), z) → oplus(*(x, z), *(y, z))
*(x, oplus(y, z)) → oplus(*(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)) → *(otimes(z0, z1), z2)
*(1, z0) → z0
*(+(z0, z1), z2) → oplus(*(z0, z2), *(z1, z2))
*(z0, oplus(z1, z2)) → oplus(*(z0, z1), *(z0, z2))

Tuples:

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(1, z0) → c1
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

S tuples:

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(1, z0) → c1
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:

*'

Compound Symbols:

c, c1, c2, c3

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

*'(1, z0) → c1

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, *(z1, z2)) → *(otimes(z0, z1), z2)
*(1, z0) → z0
*(+(z0, z1), z2) → oplus(*(z0, z2), *(z1, z2))
*(z0, oplus(z1, z2)) → oplus(*(z0, z1), *(z0, z2))

Tuples:

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

S tuples:

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:

*'

Compound Symbols:

c, c2, c3

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

*(z0, *(z1, z2)) → *(otimes(z0, z1), z2)
*(1, z0) → z0
*(+(z0, z1), z2) → oplus(*(z0, z2), *(z1, z2))
*(z0, oplus(z1, z2)) → oplus(*(z0, z1), *(z0, z2))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

S tuples:

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

*'

Compound Symbols:

c, c2, c3

(7) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

We considered the (Usable) Rules:none
And the Tuples:

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(*(x₁, x₂)) = [1] + x₂   
POL(*'(x₁, x₂)) = x₁ + x₂ + x₁·x₂   
POL(+(x₁, x₂)) = [1] + x₁ + x₂   
POL(c(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(oplus(x₁, x₂)) = [1] + x₁ + x₂   
POL(otimes(x₁, x₂)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

S tuples:none
K tuples:

*'(z0, *(z1, z2)) → c(*'(otimes(z0, z1), z2))
*'(+(z0, z1), z2) → c2(*'(z0, z2), *'(z1, z2))
*'(z0, oplus(z1, z2)) → c3(*'(z0, z1), *'(z0, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

*'

Compound Symbols:

c, c2, c3

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

The set S is empty

(10) BOUNDS(O(1), O(1))