The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(x, y), z) → +(*(x, z), *(y, z))
*(x, 1) → x
*(1, y) → y

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:

*'

Compound Symbols:

c, c1, c2, c3

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

Removed 2 trailing nodes:

*'(z0, 1) → c2
*'(1, z0) → c3

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:

*'

Compound Symbols:

c, c1

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

*'

Compound Symbols:

c, c1

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)) 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))
*'(+(z0, z1), z2) → c1(*'(z0, z2), *'(z1, z2))

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

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

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

POL(*'(x₁, x₂)) = x₁ + x₂ + x₂² + x₁²·x₂ + x₁·x₂²   
POL(+(x₁, x₂)) = [1] + x₁ + x₂   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁, x₂)) = x₁ + x₂   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

S tuples:none
K tuples:

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

Defined Rule Symbols:none

Defined Pair Symbols:

*'

Compound Symbols:

c, c1

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

The set S is empty

(10) BOUNDS(1, 1)