Correctness

Some facts:

• Algorithm maintains the invariant that for all ``outside'' nodes v, d[v] is the length of the shortest path from s to v using only inside nodes. Invariant maintained by looking at all possible extensions of ``inside'' paths.
• The distances of nodes along a shortest path from s to t is monotonically non-decreasing, so it is reasonable to grow the path in increasing order of distance.
• Let u_i be the ith node brought to the inside. The sequence d[u_i] is non decreasing: Dijkstra's sorts the nodes by distance.
• For any given node u, d[u] starts at infinity and decreases until u is brought inside.
• Induction: When u is brought inside, $d[u]=\delta(s,u)$.