### (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__f(f(a)) → c(f(g(f(a))))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c(X)) → c(X)
mark(g(X)) → g(mark(X))
a__f(X) → f(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:

a__f(f(a)) → c(f(g(f(a))))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c(X)) → c(X)
mark(g(X)) → g(mark(X))
a__f(X) → f(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: 3
Accept states: [4]
Transitions:
3→4[a__f_1|0, mark_1|0, f_1|1, a|1, c_1|1]
3→5[c_1|1]
3→9[a__f_1|1, f_1|2]
3→10[g_1|1]
3→11[c_1|2]
4→4[f_1|0, a|0, c_1|0, g_1|0]
5→6[f_1|1]
6→7[g_1|1]
7→8[f_1|1]
8→4[a|1]
9→4[mark_1|1, a|1, c_1|1]
9→9[a__f_1|1, f_1|2]
9→10[g_1|1]
9→11[c_1|2]
10→4[mark_1|1, a|1, c_1|1]
10→9[a__f_1|1, f_1|2]
10→10[g_1|1]
10→11[c_1|2]
11→12[f_1|2]
12→13[g_1|2]
13→14[f_1|2]
14→4[a|2]