(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:

a(b(x)) → b(a(x))
a(c(x)) → x

Rewrite Strategy: FULL

(1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, 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:

a(b(x)) → b(a(x))
a(c(x)) → 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: 2
Accept states: [3]
Transitions:
2→3[a_1|0, b_1|1, c_1|1]
2→4[b_1|1]
3→3[b_1|0, c_1|0]
4→3[a_1|1, b_1|1, c_1|1]
4→4[b_1|1]

(4) BOUNDS(1, n^1)