(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:
f(f(a)) → f(g(n__f(n__a)))
f(X) → n__f(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__a) → a
activate(X) → X
Rewrite Strategy: INNERMOST
(1) 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 3.
The certificate found is represented by the following graph.
Start state: 7
Accept states: [8]
Transitions:
7→8[f_1|0, a|0, activate_1|0, n__f_1|1, n__a|1, a|1, g_1|1, n__a|2]
7→9[f_1|1, n__f_1|2]
7→10[f_1|2, n__f_1|3]
8→8[g_1|0, n__f_1|0, n__a|0]
9→8[activate_1|1, n__f_1|1, a|1, n__a|1, g_1|1, n__a|2]
9→9[f_1|1, n__f_1|2]
9→10[f_1|2, n__f_1|3]
10→11[g_1|2]
11→12[n__f_1|2]
12→8[n__a|2]