(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
isEmpty(empty) → true
isEmpty(edge(x, y, i)) → false
from(edge(x, y, i)) → x
to(edge(x, y, i)) → y
rest(edge(x, y, i)) → i
rest(empty) → empty
reach(x, y, i, h) → if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
if1(true, b1, b2, b3, x, y, i, h) → true
if1(false, b1, b2, b3, x, y, i, h) → if2(b1, b2, b3, x, y, i, h)
if2(true, b2, b3, x, y, i, h) → false
if2(false, b2, b3, x, y, i, h) → if3(b2, b3, x, y, i, h)
if3(false, b3, x, y, i, h) → reach(x, y, rest(i), edge(from(i), to(i), h))
if3(true, b3, x, y, i, h) → if4(b3, x, y, i, h)
if4(true, x, y, i, h) → true
if4(false, x, y, i, h) → or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
eq(s(x), s(y)) →+ eq(x, y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(x), y / s(y)].
The result substitution is [ ].
(2) BOUNDS(n^1, INF)