### (0) Obligation:

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

The TRS R consists of the following rules:

*(x, +(y, z)) → +(*(x, y), *(x, z))

Rewrite Strategy: FULL

### (1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

### (2) Obligation:

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

The TRS R consists of the following rules:

*(x, +(y, z)) → +(*(x, y), *(x, z))

Rewrite Strategy: INNERMOST

### (3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
Tuples:

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

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:

*'

Compound Symbols:

c

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

*'

Compound Symbols:

c

### (7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) 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))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(*'(x1, x2)) = [2] + [2]x2
POL(+(x1, x2)) = [2] + x1 + x2
POL(c(x1, x2)) = x1 + x2

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
S tuples:none
K tuples:

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

Defined Pair Symbols:

*'

Compound Symbols:

c

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

The set S is empty