(0) Obligation:
The Runtime Complexity (full) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
f(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(0)) → 0
Rewrite Strategy: FULL
(1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS is a non-duplicating overlay system, we have rc = irc.
(2) 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(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(0)) → 0
Rewrite Strategy: INNERMOST
(3) 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: 3
Accept states: [4]
Transitions:
3→4[f_1|0, p_1|0, 0|1]
3→5[cons_1|1]
3→6[f_1|1]
3→9[cons_1|2]
4→4[0|0, cons_1|0, s_1|0]
5→4[0|1]
6→7[p_1|1]
6→4[0|2]
7→8[s_1|1]
8→4[0|1]
9→4[0|2]
(4) BOUNDS(1, n^1)