(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(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(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, g_1|1]
1→3[g_1|1]
1→5[a__f_1|1, f_1|2]
1→6[h_1|1]
1→7[g_1|2]
2→2[g_1|0, h_1|0, f_1|0]
3→4[h_1|1]
4→2[f_1|1]
5→2[mark_1|1, g_1|1]
5→5[a__f_1|1, f_1|2]
5→6[h_1|1]
5→7[g_1|2]
6→2[mark_1|1, g_1|1]
6→5[a__f_1|1, f_1|2]
6→6[h_1|1]
6→7[g_1|2]
7→8[h_1|2]
8→5[f_1|2]