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