(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)) → c(n__f(n__g(n__f(n__a))))
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(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 2.
The certificate found is represented by the following graph.
Start state: 9
Accept states: [10]
Transitions:
9→10[f_1|0, g_1|0, a|0, activate_1|0, n__f_1|1, n__g_1|1, n__a|1, a|1, c_1|1, n__a|2]
9→11[f_1|1, n__f_1|2]
9→12[g_1|1, n__g_1|2]
9→13[c_1|2]
10→10[c_1|0, n__f_1|0, n__g_1|0, n__a|0]
11→10[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]
11→11[f_1|1, n__f_1|2]
11→12[g_1|1, n__g_1|2]
11→13[c_1|2]
12→10[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]
12→11[f_1|1, n__f_1|2]
12→12[g_1|1, n__g_1|2]
12→13[c_1|2]
13→14[n__f_1|2]
14→15[n__g_1|2]
15→16[n__f_1|2]
16→10[n__a|2]