(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))
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:
g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))
S is empty.
Rewrite Strategy: FULL
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
g/1
h/1
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
g(f(x)) → f(h(x))
h(x) → g(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
g(f(x)) →+ f(g(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / f(x)].
The result substitution is [ ].
(6) BOUNDS(n^1, INF)