### (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), *(x, z)) → *(x, +(y, z))

+(+(x, y), z) → +(x, +(y, z))

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

Rewrite Strategy: FULL

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

The TRS does not nest defined symbols.

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:

+(+(x, y), z) → +(x, +(y, z))

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

### (2) 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), *(x, z)) → *(x, +(y, z))

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

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

### (4) 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), *(x, z)) → *(x, +(y, z))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT
### (6) Obligation:

Complexity Dependency Tuples Problem

Rules:

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

Tuples:

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

S tuples:

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

K tuples:none

Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

### (8) Obligation:

Complexity Dependency Tuples Problem

Rules:none

Tuples:

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

S tuples:

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

K tuples:none

Defined Rule Symbols:none

Defined Pair Symbols:

+'

Compound Symbols:

c

### (9) 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), *(z0, z2)) → c(+'(z1, z2))

We considered the (Usable) Rules:none

And the Tuples:

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

The order we found is given by the following interpretation:

Polynomial interpretation :

POL(***(x**_{1}, x_{2})) = [1] + x_{2}^{ } _{ }

POL(**+'(x**_{1}, x_{2})) = x_{2}^{ } _{ }

POL(**c(x**_{1})) = x_{1}^{ } _{ }

### (10) Obligation:

Complexity Dependency Tuples Problem

Rules:none

Tuples:

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

S tuples:none

K tuples:

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

Defined Rule Symbols:none

Defined Pair Symbols:

+'

Compound Symbols:

c

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

The set S is empty

### (12) BOUNDS(1, 1)