(0) 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:
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: INNERMOST
(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 (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
(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 4.
The certificate found is represented by the following graph.
Start state: 6
Accept states: [7]
Transitions:
6→7[g_1|0, top_1|0, proper_1|0, active_1|0, h_1|0]
6→8[ok_1|1]
6→9[top_1|1]
6→10[top_1|1]
6→11[ok_1|1]
6→12[ok_1|1]
6→13[mark_1|1]
6→14[ok_1|1]
6→15[top_1|2]
6→16[top_1|2]
6→19[top_1|3]
6→20[top_1|3]
6→22[top_1|4]
7→7[ok_1|0, c|0, d|0, mark_1|0]
8→7[g_1|1]
8→8[ok_1|1]
9→7[active_1|1]
9→13[mark_1|1]
10→7[proper_1|1]
10→11[ok_1|1]
10→12[ok_1|1]
11→7[c|1]
12→7[d|1]
13→7[d|1]
14→7[h_1|1]
14→14[ok_1|1]
15→13[proper_1|2]
15→17[ok_1|2]
16→11[active_1|2]
16→12[active_1|2]
16→18[mark_1|2]
17→7[d|2]
18→7[d|2]
19→17[active_1|3]
20→18[proper_1|3]
20→21[ok_1|3]
21→7[d|3]
22→21[active_1|4]
(4) BOUNDS(1, n^1)