(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(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: 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(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 (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
(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 1.
The certificate found is represented by the following graph.
Start state: 7
Accept states: [8]
Transitions:
7→8[g_1|0, top_1|0, f_1|0, c_1|0, d_1|0, h_1|0]
7→9[ok_1|1]
7→10[top_1|1]
7→11[top_1|1]
7→12[mark_1|1]
7→13[ok_1|1]
7→14[ok_1|1]
7→15[ok_1|1]
7→16[mark_1|1]
7→17[ok_1|1]
8→8[ok_1|0, active_1|0, mark_1|0, proper_1|0]
9→8[g_1|1]
9→9[ok_1|1]
10→8[active_1|1]
11→8[proper_1|1]
12→8[f_1|1]
12→12[mark_1|1]
12→13[ok_1|1]
13→8[f_1|1]
13→12[mark_1|1]
13→13[ok_1|1]
14→8[c_1|1]
14→14[ok_1|1]
15→8[d_1|1]
15→15[ok_1|1]
16→8[h_1|1]
16→16[mark_1|1]
16→17[ok_1|1]
17→8[h_1|1]
17→16[mark_1|1]
17→17[ok_1|1]
(4) BOUNDS(1, n^1)