(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:
a__f(f(a)) → c(f(g(f(a))))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c(X)) → c(X)
mark(g(X)) → g(mark(X))
a__f(X) → f(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: 1
Accept states: [2]
Transitions:
1→2[a__f_1|0, mark_1|0, f_1|1, a|1, c_1|1]
1→3[c_1|1]
1→7[a__f_1|1, f_1|2]
1→8[g_1|1]
1→9[c_1|2]
2→2[f_1|0, a|0, c_1|0, g_1|0]
3→4[f_1|1]
4→5[g_1|1]
5→6[f_1|1]
6→2[a|1]
7→2[mark_1|1, a|1, c_1|1]
7→7[a__f_1|1, f_1|2]
7→8[g_1|1]
7→9[c_1|2]
8→2[mark_1|1, a|1, c_1|1]
8→7[a__f_1|1, f_1|2]
8→8[g_1|1]
8→9[c_1|2]
9→10[f_1|2]
10→11[g_1|2]
11→12[f_1|2]
12→2[a|2]