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, 0) → x
+(x, i(x)) → 0
+(+(x, y), z) → +(x, +(y, z))
*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(x, y), z) → +(*(x, z), *(y, z))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(z0, 0) → z0
+(z0, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))

Tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

S tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

K tuples:none
Defined Rule Symbols:

+, *

Defined Pair Symbols:

+', *'

Compound Symbols:

c2, c3, c4

(3) CdtPolyRedPairProof (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)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

We considered the (Usable) Rules:

*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))
+(z0, 0) → z0
+(z0, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))

And the Tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

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

POL(*(x₁, x₂)) = [1] + [2]x₂ + [3]x₂²   
POL(*'(x₁, x₂)) = [2] + x₁ + x₂ + [2]x₁·x₂   
POL(+(x₁, x₂)) = [3] + [2]x₁ + [3]x₂ + [3]x₂² + x₁·x₂ + [3]x₁²   
POL(+'(x₁, x₂)) = 0   
POL(0) = 0   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(i(x₁)) = 0   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(z0, 0) → z0
+(z0, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))

Tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

S tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))

K tuples:

*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

Defined Rule Symbols:

+, *

Defined Pair Symbols:

+', *'

Compound Symbols:

c2, c3, c4

(5) CdtPolyRedPairProof (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) → c2(+'(z0, +(z1, z2)), +'(z1, z2))

We considered the (Usable) Rules:

*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))
+(z0, 0) → z0
+(z0, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))

And the Tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

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

POL(*(x₁, x₂)) = x₁ + x₂ + [3]x₁·x₂   
POL(*'(x₁, x₂)) = [1] + x₁ + x₂ + [3]x₁·x₂   
POL(+(x₁, x₂)) = [1] + [3]x₁ + x₂   
POL(+'(x₁, x₂)) = [2]x₁   
POL(0) = [3]   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(i(x₁)) = [3]   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(z0, 0) → z0
+(z0, i(z0)) → 0
+(+(z0, z1), z2) → +(z0, +(z1, z2))
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))

Tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))

S tuples:none
K tuples:

*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))

Defined Rule Symbols:

+, *

Defined Pair Symbols:

+', *'

Compound Symbols:

c2, c3, c4

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))