### (0) Obligation:

Runtime Complexity TRS:
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(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

*'(+(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 + [3]x1·x2
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:

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
K tuples:

*'(+(z0, z1), z2) → c1(*'(z0, z2), *'(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

*'

Compound Symbols:

c, c1

### (9) CdtRuleRemovalProof (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)) → c(*'(z0, z1), *'(z0, 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)) = [2] + x1 + [2]x2 + [2]x1·x2
POL(+(x1, x2)) = [2] + x1 + x2
POL(c(x1, x2)) = x1 + x2
POL(c1(x1, x2)) = x1 + x2

### (10) 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) → c1(*'(z0, z2), *'(z1, z2))
*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

*'

Compound Symbols:

c, c1

### (11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty