(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