### (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(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Rewrite Strategy: FULL

### (1) 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.

### (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:

f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → 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 2.
The certificate found is represented by the following graph.
Start state: 4
Accept states: [5]
Transitions:
4→5[f_1|0, h_1|0, activate_1|0, n__f_1|1, n__h_1|1, g_1|1]
4→6[g_1|1]
4→8[h_1|1, n__h_1|2]
4→9[f_1|1, n__f_1|2]
4→10[g_1|2]
5→5[g_1|0, n__h_1|0, n__f_1|0]
6→7[n__h_1|1]
7→5[n__f_1|1]
8→5[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]
8→8[h_1|1, n__h_1|2]
8→9[f_1|1, n__f_1|2]
8→10[g_1|2]
9→5[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]
9→8[h_1|1, n__h_1|2]
9→9[f_1|1, n__f_1|2]
9→10[g_1|2]
10→11[n__h_1|2]
11→9[n__f_1|2]