### (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) → f(c(a))

f(c(X)) → X

f(c(a)) → f(d(b))

f(a) → f(d(a))

f(d(X)) → X

f(c(b)) → f(d(a))

e(g(X)) → e(X)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, 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(a) → f(c(a))

f(c(X)) → X

f(c(a)) → f(d(b))

f(a) → f(d(a))

f(d(X)) → X

f(c(b)) → f(d(a))

e(g(X)) → e(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 3.

The certificate found is represented by the following graph.

Start state: 3

Accept states: [4]

Transitions:

3→4[f_1|0, e_1|0, a|1, c_1|1, d_1|1, b|1, g_1|1, e_1|1, a|2, b|2, b|3]

3→5[f_1|1]

3→7[f_1|1]

3→9[f_1|1]

3→11[f_1|2]

4→4[a|0, c_1|0, d_1|0, b|0, g_1|0]

5→6[c_1|1]

6→4[a|1]

7→8[d_1|1]

8→4[a|1]

9→10[d_1|1]

10→4[b|1]

11→12[d_1|2]

12→4[b|2]

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