(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(f(g(f(a))))
active(f(X)) → f(active(X))
f(mark(X)) → mark(f(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(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(f(g(f(a))))
active(f(X)) → f(active(X))
proper(f(X)) → f(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))
f(ok(X)) → ok(f(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: 5
Accept states: [6]
Transitions:
5→6[g_1|0, top_1|0, f_1|0, proper_1|0]
5→7[ok_1|1]
5→8[top_1|1]
5→9[top_1|1]
5→10[mark_1|1]
5→11[ok_1|1]
5→12[ok_1|1]
5→13[top_1|2]
6→6[ok_1|0, active_1|0, mark_1|0, a|0]
7→6[g_1|1]
7→7[ok_1|1]
8→6[active_1|1]
9→6[proper_1|1]
9→12[ok_1|1]
10→6[f_1|1]
10→10[mark_1|1]
10→11[ok_1|1]
11→6[f_1|1]
11→10[mark_1|1]
11→11[ok_1|1]
12→6[a|1]
13→12[active_1|2]
(4) BOUNDS(1, n^1)