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(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(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(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(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))
f(mark(X)) → mark(f(X))
c(ok(X)) → ok(c(X))
f(ok(X)) → ok(f(X))
d(ok(X)) → ok(d(X))
top(mark(X)) → top(proper(X))
h(mark(X)) → mark(h(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 does not nest defined symbols, 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))
c(ok(X)) → ok(c(X))
f(ok(X)) → ok(f(X))
d(ok(X)) → ok(d(X))
top(mark(X)) → top(proper(X))
h(mark(X)) → mark(h(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 1.
The certificate found is represented by the following graph.

Start state: 13
Accept states: [14]
Transitions:
13→14[g_1|0, top_1|0, f_1|0, c_1|0, d_1|0, h_1|0]
13→15[ok_1|1]
13→16[top_1|1]
13→17[top_1|1]
13→18[mark_1|1]
13→19[ok_1|1]
13→20[ok_1|1]
13→21[ok_1|1]
13→22[mark_1|1]
13→23[ok_1|1]
14→14[ok_1|0, active_1|0, mark_1|0, proper_1|0]
15→14[g_1|1]
15→15[ok_1|1]
16→14[active_1|1]
17→14[proper_1|1]
18→14[f_1|1]
18→18[mark_1|1]
18→19[ok_1|1]
19→14[f_1|1]
19→18[mark_1|1]
19→19[ok_1|1]
20→14[c_1|1]
20→20[ok_1|1]
21→14[d_1|1]
21→21[ok_1|1]
22→14[h_1|1]
22→22[mark_1|1]
22→23[ok_1|1]
23→14[h_1|1]
23→22[mark_1|1]
23→23[ok_1|1]

(6) BOUNDS(1, n^1)