(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(a))) → mark(c(f(g(f(a)))))
active(f(X)) → f(active(X))
active(g(X)) → g(active(X))
f(mark(X)) → mark(f(X))
g(mark(X)) → mark(g(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(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(a))) → mark(c(f(g(f(a)))))
active(f(X)) → f(active(X))
active(g(X)) → g(active(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(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))
g(mark(X)) → mark(g(X))
top(mark(X)) → top(proper(X))
proper(a) → ok(a)
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 2.
The certificate found is represented by the following graph.
Start state: 6
Accept states: [7]
Transitions:
6→7[g_1|0, top_1|0, f_1|0, c_1|0, proper_1|0]
6→8[ok_1|1]
6→9[mark_1|1]
6→10[top_1|1]
6→11[top_1|1]
6→12[mark_1|1]
6→13[ok_1|1]
6→14[ok_1|1]
6→15[ok_1|1]
6→16[top_1|2]
7→7[ok_1|0, active_1|0, mark_1|0, a|0]
8→7[g_1|1]
8→8[ok_1|1]
8→9[mark_1|1]
9→7[g_1|1]
9→8[ok_1|1]
9→9[mark_1|1]
10→7[active_1|1]
11→7[proper_1|1]
11→15[ok_1|1]
12→7[f_1|1]
12→12[mark_1|1]
12→13[ok_1|1]
13→7[f_1|1]
13→12[mark_1|1]
13→13[ok_1|1]
14→7[c_1|1]
14→14[ok_1|1]
15→7[a|1]
16→15[active_1|2]
(4) BOUNDS(1, n^1)