### (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(a) → g(h(a))

h(g(x)) → g(h(f(x)))

k(x, h(x), a) → h(x)

k(f(x), y, x) → f(x)

Rewrite Strategy: FULL

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

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

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

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

k(x, h(x), a) → h(x)

k(f(x), y, 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:

h(g(x)) → g(h(f(x)))

f(a) → g(h(a))

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:

h(g(x)) → g(h(f(x)))

f(a) → g(h(a))

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[h_1|0, f_1|0]

5→7[g_1|1]

5→9[g_1|1]

6→6[g_1|0, a|0]

7→8[h_1|1]

7→11[g_1|2]

8→6[f_1|1]

8→9[g_1|1]

9→10[h_1|1]

10→6[a|1]

11→12[h_1|2]

12→9[f_1|2]

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