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:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(-(neg(z0), neg(z0)), -(neg(z1), neg(z1))) → -(-(z0, z1), -(z0, z1))

Tuples:

-'(-(neg(z0), neg(z0)), -(neg(z1), neg(z1))) → c(-'(-(z0, z1), -(z0, z1)), -'(z0, z1), -'(z0, z1))

S tuples:

-'(-(neg(z0), neg(z0)), -(neg(z1), neg(z1))) → c(-'(-(z0, z1), -(z0, z1)), -'(z0, z1), -'(z0, z1))

K tuples:none
Defined Rule Symbols:

-

Defined Pair Symbols:

-'

Compound Symbols:

c

(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.

-'(-(neg(z0), neg(z0)), -(neg(z1), neg(z1))) → c(-'(-(z0, z1), -(z0, z1)), -'(z0, z1), -'(z0, z1))

We considered the (Usable) Rules:

-(-(neg(z0), neg(z0)), -(neg(z1), neg(z1))) → -(-(z0, z1), -(z0, z1))

And the Tuples:

-'(-(neg(z0), neg(z0)), -(neg(z1), neg(z1))) → c(-'(-(z0, z1), -(z0, z1)), -'(z0, z1), -'(z0, z1))

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

POL(-(x₁, x₂)) = [2]x₁²   
POL(-'(x₁, x₂)) = [3] + [2]x₁²   
POL(c(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(neg(x₁)) = [2] + x₁   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(-(neg(z0), neg(z0)), -(neg(z1), neg(z1))) → -(-(z0, z1), -(z0, z1))

Tuples:

-'(-(neg(z0), neg(z0)), -(neg(z1), neg(z1))) → c(-'(-(z0, z1), -(z0, z1)), -'(z0, z1), -'(z0, z1))

S tuples:none
K tuples:

-'(-(neg(z0), neg(z0)), -(neg(z1), neg(z1))) → c(-'(-(z0, z1), -(z0, z1)), -'(z0, z1), -'(z0, z1))

Defined Rule Symbols:

-

Defined Pair Symbols:

-'

Compound Symbols:

c

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))