### (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:

active(f(f(a))) → mark(f(g(f(a))))

active(f(X)) → f(active(X))

f(mark(X)) → mark(f(X))

proper(f(X)) → f(proper(X))

proper(a) → ok(a)

proper(g(X)) → g(proper(X))

f(ok(X)) → ok(f(X))

g(ok(X)) → ok(g(X))

top(mark(X)) → top(proper(X))

top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

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

The following defined symbols can occur below the 0th argument of top: proper, active

The following defined symbols can occur below the 0th argument of proper: proper, active

The following defined symbols can occur below the 0th argument of active: proper, active

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

active(f(f(a))) → mark(f(g(f(a))))

active(f(X)) → f(active(X))

proper(f(X)) → f(proper(X))

proper(g(X)) → g(proper(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(ok(X)) → ok(g(X))

top(ok(X)) → top(active(X))

f(mark(X)) → mark(f(X))

f(ok(X)) → ok(f(X))

top(mark(X)) → top(proper(X))

proper(a) → ok(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:

g(ok(X)) → ok(g(X))

top(ok(X)) → top(active(X))

f(mark(X)) → mark(f(X))

f(ok(X)) → ok(f(X))

top(mark(X)) → top(proper(X))

proper(a) → ok(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: 9

Accept states: [10]

Transitions:

9→10[g_1|0, top_1|0, f_1|0, proper_1|0]

9→11[ok_1|1]

9→12[top_1|1]

9→13[top_1|1]

9→14[mark_1|1]

9→15[ok_1|1]

9→16[ok_1|1]

9→17[top_1|2]

10→10[ok_1|0, active_1|0, mark_1|0, a|0]

11→10[g_1|1]

11→11[ok_1|1]

12→10[active_1|1]

13→10[proper_1|1]

13→16[ok_1|1]

14→10[f_1|1]

14→14[mark_1|1]

14→15[ok_1|1]

15→10[f_1|1]

15→14[mark_1|1]

15→15[ok_1|1]

16→10[a|1]

17→16[active_1|2]

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