(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(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
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 4.
The certificate found is represented by the following graph.
Start state: 13
Accept states: [14]
Transitions:
13→14[f_1|0, c_1|0, h_1|0, g_1|0, d_1|0, activate_1|0, n__f_1|1, n__g_1|1, n__d_1|1, g_1|1, d_1|1, n__g_1|2, n__d_1|2]
13→15[d_1|1, n__d_1|2]
13→16[c_1|1]
13→17[f_1|1, n__f_1|2]
13→18[d_1|2, n__d_1|3]
13→19[c_1|2]
13→22[d_1|3, n__d_1|4]
14→14[n__f_1|0, n__g_1|0, n__d_1|0]
15→14[activate_1|1, n__f_1|1, g_1|1, n__g_1|1, d_1|1, n__d_1|1, n__g_1|2, n__d_1|2]
15→17[f_1|1, n__f_1|2]
15→19[c_1|2]
15→22[d_1|3, n__d_1|4]
16→14[n__d_1|1]
17→14[activate_1|1, n__f_1|1, g_1|1, n__g_1|1, d_1|1, n__d_1|1, n__g_1|2, n__d_1|2]
17→17[f_1|1, n__f_1|2]
17→19[c_1|2]
17→22[d_1|3, n__d_1|4]
18→16[activate_1|2]
18→14[d_1|2, n__d_1|2, n__d_1|3]
19→20[n__f_1|2]
20→21[n__g_1|2]
21→17[n__f_1|2]
22→19[activate_1|3]
22→23[f_1|3, n__f_1|4]
22→20[n__f_1|3]
23→20[activate_1|3]
23→21[g_1|3, n__g_1|3, n__g_1|4]