(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0, x2) → x2
f(x1, 0) → g(x1, 0)
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:
g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0, x2) → x2
f(x1, 0) → g(x1, 0)
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
g(S(x), y) →+ g(x, S(y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / S(x)].
The result substitution is [y / S(y)].
(4) BOUNDS(n^1, INF)