The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(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(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(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(g(X)) → mark(h(X))
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(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))
proper(c) → ok(c)
proper(d) → ok(d)
active(c) → mark(d)
top(mark(X)) → top(proper(X))
h(ok(X)) → ok(h(X))

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))
proper(c) → ok(c)
proper(d) → ok(d)
active(c) → mark(d)
top(mark(X)) → top(proper(X))
h(ok(X)) → ok(h(X))

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 4.
The certificate found is represented by the following graph.

Start state: 11
Accept states: [12]
Transitions:
11→12[g_1|0, top_1|0, proper_1|0, active_1|0, h_1|0]
11→13[ok_1|1]
11→14[top_1|1]
11→15[top_1|1]
11→16[ok_1|1]
11→17[ok_1|1]
11→18[mark_1|1]
11→19[ok_1|1]
11→20[top_1|2]
11→21[top_1|2]
11→24[top_1|3]
11→25[top_1|3]
11→27[top_1|4]
12→12[ok_1|0, c|0, d|0, mark_1|0]
13→12[g_1|1]
13→13[ok_1|1]
14→12[active_1|1]
14→18[mark_1|1]
15→12[proper_1|1]
15→16[ok_1|1]
15→17[ok_1|1]
16→12[c|1]
17→12[d|1]
18→12[d|1]
19→12[h_1|1]
19→19[ok_1|1]
20→18[proper_1|2]
20→22[ok_1|2]
21→16[active_1|2]
21→17[active_1|2]
21→23[mark_1|2]
22→12[d|2]
23→12[d|2]
24→22[active_1|3]
25→23[proper_1|3]
25→26[ok_1|3]
26→12[d|3]
27→26[active_1|4]

(6) BOUNDS(1, n^1)