### (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(f(x)) → f(x)

f(s(x)) → f(x)

g(s(0)) → g(f(s(0)))

Rewrite Strategy: FULL

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

The following defined symbols can occur below the 0th argument of g: f

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

f(f(x)) → f(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(0)) → g(f(s(0)))

f(s(x)) → f(x)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, 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(0)) → g(f(s(0)))

f(s(x)) → f(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 2.

The certificate found is represented by the following graph.

Start state: 5

Accept states: [6]

Transitions:

5→6[g_1|0, f_1|0, f_1|1]

5→7[g_1|1]

6→6[s_1|0, 0|0]

7→8[f_1|1]

7→9[f_1|2]

8→9[s_1|1]

9→6[0|1]

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