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