(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
Rewrite Strategy: FULL
(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:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
S is empty.
Rewrite Strategy: FULL
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
+/1
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x)))
S is empty.
Rewrite Strategy: FULL
(5) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
sum(s(x)) →+ +(sum(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(x)].
The result substitution is [ ].
(6) BOUNDS(n^1, INF)