### (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:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

Rewrite Strategy: FULL

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

The following defined symbols can occur below the 1th argument of plus: times

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))

### (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:

times(X, s(Y)) → plus(X, times(Y, X))

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:

times(X, s(Y)) → plus(X, times(Y, X))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, s(z1)) → plus(z0, times(z1, z0))
Tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))
S tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))
K tuples:none
Defined Rule Symbols:

times

Defined Pair Symbols:

TIMES

Compound Symbols:

c

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

times(z0, s(z1)) → plus(z0, times(z1, z0))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))
S tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

TIMES

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.

TIMES(z0, s(z1)) → c(TIMES(z1, z0))
We considered the (Usable) Rules:none
And the Tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(TIMES(x1, x2)) = x1 + x2
POL(c(x1)) = x1
POL(s(x1)) =  + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))
S tuples:none
K tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))
Defined Rule Symbols:none

Defined Pair Symbols:

TIMES

Compound Symbols:

c

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

The set S is empty