(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:
f(f(x)) → f(x)
f(s(x)) → f(x)
g(s(0)) → g(f(s(0)))
Rewrite Strategy: INNERMOST
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 0th argument of g: f
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(f(x)) → f(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:
g(s(0)) → g(f(s(0)))
f(s(x)) → f(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 2.
The certificate found is represented by the following graph.
Start state: 3
Accept states: [4]
Transitions:
3→4[g_1|0, f_1|0, f_1|1]
3→5[g_1|1]
4→4[s_1|0, 0|0]
5→6[f_1|1]
5→7[f_1|2]
6→7[s_1|1]
7→4[0|1]
(4) BOUNDS(1, n^1)