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

(0) Obligation:

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

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

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

(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), *(z0, z2)) → c(+'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))

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

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

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

POL(*(x₁, x₂)) = [1] + x₂   
POL(+(x₁, x₂)) = [5] + [4]x₁ + [4]x₂   
POL(+'(x₁, x₂)) = [5] + x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(u) = [2]   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))

Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

(7) 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) → c1(+'(z0, +(z1, z2)), +'(z1, z2))

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

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

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

POL(*(x₁, x₂)) = x₁ + x₂   
POL(+(x₁, x₂)) = [4] + x₁ + x₂   
POL(+'(x₁, x₂)) = [5] + [4]x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(u) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:none
K tuples:

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

Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

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

The set S is empty

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