### (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:

f(g(x), s(0), y) → f(g(s(0)), y, g(x))

g(s(x)) → s(g(x))

g(0) → 0

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:

f(g(x), s(0), y) → f(g(s(0)), y, g(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:

g(s(x)) → s(g(x))

g(0) → 0

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:

g(s(x)) → s(g(x))

g(0) → 0

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: 3

Accept states: [4]

Transitions:

3→4[g_1|0, 0|1]

3→5[s_1|1]

4→4[s_1|0, 0|0]

5→4[g_1|1, 0|1]

5→5[s_1|1]

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