### (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:

w(r(x)) → r(w(x))

b(r(x)) → r(b(x))

b(w(x)) → w(b(x))

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:

b(w(x)) → w(b(x))

### (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:

b(r(x)) → r(b(x))

w(r(x)) → r(w(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:

b(r(x)) → r(b(x))

w(r(x)) → r(w(x))

Rewrite Strategy: INNERMOST

### (5) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.

The certificate found is represented by the following graph.

Start state: 5

Accept states: [6]

Transitions:

5→6[b_1|0, w_1|0]

5→7[r_1|1]

5→8[r_1|1]

6→6[r_1|0]

7→6[b_1|1]

7→7[r_1|1]

8→6[w_1|1]

8→8[r_1|1]

### (6) BOUNDS(1, n^1)