(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, 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:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))
S is empty.
Rewrite Strategy: FULL
(3) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
ackin(s(X), s(Y)) →+ u21(ackin(s(X), Y), X)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [Y / s(Y)].
The result substitution is [ ].
(4) BOUNDS(n^1, INF)