(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
dbl(S(0), S(0)) → S(S(S(S(0))))
save(S(x)) → dbl(0, save(x))
save(0) → 0
dbl(0, y) → y
Rewrite Strategy: INNERMOST
(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)
Transformed TRS to relative TRS where S is empty.
(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
dbl(S(0), S(0)) → S(S(S(S(0))))
save(S(x)) → dbl(0, save(x))
save(0) → 0
dbl(0, y) → y
S is empty.
Rewrite Strategy: INNERMOST
(3) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
save(S(x)) →+ dbl(0, save(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / S(x)].
The result substitution is [ ].
(4) BOUNDS(n^1, INF)