### (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:

a(b(x)) → b(b(a(x)))

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:

a(b(x)) → b(b(a(x)))

Rewrite Strategy: INNERMOST

### (3) 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: 2

Accept states: [3]

Transitions:

2→3[a_1|0]

2→4[b_1|1]

3→3[b_1|0]

4→5[b_1|1]

5→3[a_1|1]

5→4[b_1|1]

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