(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:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))
Rewrite Strategy: INNERMOST
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
b(w(x)) → w(b(x))
(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:
b(r(x)) → r(b(x))
w(r(x)) → r(w(x))
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 1.
The certificate found is represented by the following graph.
Start state: 3
Accept states: [4]
Transitions:
3→4[b_1|0, w_1|0]
3→5[r_1|1]
3→6[r_1|1]
4→4[r_1|0]
5→4[b_1|1]
5→5[r_1|1]
6→4[w_1|1]
6→6[r_1|1]
(4) BOUNDS(1, n^1)