(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(x, y) → g(x, y)
g(h(x), y) → h(f(x, y))
g(h(x), y) → h(g(x, 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:
f(x, y) → g(x, y)
g(h(x), y) → h(f(x, y))
g(h(x), y) → h(g(x, y))
S is empty.
Rewrite Strategy: FULL
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
f/1
g/1
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(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
f(h(x2_0)) →+ h(f(x2_0))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x2_0 / h(x2_0)].
The result substitution is [ ].
(6) BOUNDS(n^1, INF)